你不要害怕英文为什么是Don't be afraid of you
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时间：2018-07-16 03:45
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求翻译：但你是一位非常善良的人，因为你会对我说，不要害怕你的纹身，你将继续教我学习英语。是什么意思？
但你是一位非常善良的人，因为你会对我说，不要害怕你的纹身，你将继续教我学习英语。
问题补充：
But you are a very kind person, because you will say to me, do not be afraid your tattoo, you will continue to teach me to learn English.
But you are a very good man, because you will say to me, don't be afraid of your tattoo, you will continue to teach me English.
But you are an extremely good person, because you can say to me, do not have to be afraid your model community, you will continue to teach me to study English.
But you are a very kind person, because you said to me, don't be afraid of your tattoo, you will continue to teach me English.
正在翻译，请等待...
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请输入您需要翻译的文本！根据不同的作品，对于恶魔或魔鬼的定义可能是不同的。下面是某一作品中对恶魔和魔鬼的描述：&br&&br&&b&恶魔：混乱邪恶的代表&/b&&br&居住在深渊，有许多种族，不同的深渊位面往往有不同的特有的恶魔，它们的身体结构截然不同，也有不同的能力，但是它们大多都有超越人类的肉搏能力，并能使用各种类法术。&br&恶魔的信徒用死者的鲜血，作为对恶魔的献礼，鲜血让恶魔获得畏惧和服从。&br&&br&&b&1.邪恶&/b&&br&恶魔暴躁易怒、满怀恶意、独断暴力、毫无道德感而且无法预料，它们致力于破坏和毁灭一切事物（包括同类），并以此为毕生乐趣，它们经常入侵其他位面，带去毁灭与恐惧。&br&&br&&b&2.混乱&/b&&br&相对于守序的魔鬼，&b&住在无尽深渊中的恶魔们，则掌握了一切事物的本质——混乱。&/b&（熵增原理？）&br&&br&它们抗拒秩序，即使是深渊领主也无法让它们完全井井有条的做事。除非它们被法术控制，否则它们绝不可能团结，也不可能有计划的行动，上一刻的想法，下一刻就会被推翻。它们就是强大而邪恶的疯子。至于这帮家伙混乱到什么程度，看他们的老大、深渊意志的代表——深渊之主的表现就知道了：&br&&br&深渊之主， 一只无法用言语描述的存在，之所以无法描述，在于它太过混乱，混乱到连自身真名都没有，完全得不可名状……
如果说“地狱之主”代表着因为某种目的、某种**而犯下的罪恶，那祂就是毫无目的，单纯为了杀戮和毁灭而杀戮和毁灭。“深渊之主”是太古龙们为祂起得代号，便于称呼这样一位存在。&br&&br&“深渊之主”这位存在是彻头彻尾的疯子，谁也不知道祂下一步想做什么，因为连祂自己也不知道！它随时可能玩一把“自爆”，然后再花费漫长的时光从深渊里恢复——祂从来不会衡量得失之间的比例。 所以从来没有人去猜测它的想法，因为要是猜得中，恐怕那人也会怀疑自己已经被深渊气息侵染，开始丢掉引以为豪的智慧和冷静思考能力了，毕竟，“深渊之主”的世界，正常人永远不懂。&br&&br&老大都这德行，下面的就算脑袋稍微正常点，估计也好不到哪里去！（我一直深深地怀疑这帮脑残家伙为什么到现在还没被狡猾的魔鬼们搞死·····）&br&&br& ------------------------------------------------------------------------------------------------&br&&br&&b&魔鬼：守序邪恶的的代表，伪善的代名词&/b&&br&是区别与恶魔的另一种邪恶异界生物，这些守序而邪恶的生物居住于地狱之中，共有7位魔鬼统领，他们也被称为“地狱七君主”或“七撒旦（神之敌）”&br&&br&&b&1.伪善&/b&&br&什么是伪善？&br&“&b&伪善是邪恶向美德所表达的最崇高的敬意&/b&。”魔鬼如是说。&br& 残忍的恶魔？那只是凶兽罢了，&b&魔鬼&/b&&b&从来不屑于低级的杀戮，玩弄人心才是魔鬼的特长&/b&，否则的话，七宗罪里怎么没有“杀戮，残忍”这样的罪？因为那只是人性的阴暗面，不是魔鬼的赠礼。&br&&br&&b&2.守序&/b&&br&&b&地狱的法则是什么？守序的邪恶！&/b&&br&与混乱、善变的恶魔不同，魔鬼是来地狱的生物。目前魔鬼中数量最多的是巴特祖族，他们以强大的力量、邪恶的性格、无情但有效率的组织而恶名昭彰。巴特祖族有严谨的社会 街机系统，其中的全力不仅取决于力量，还取决于地位高低。&br&&br&&b&3.交易与契约&/b&&br&可参考&a href=&http://www.zhihu.com/question//answer/& class=&internal&&有人出卖灵魂给撒旦吗？过程是怎样的？ - 知乎用户的回答&/a&&br&魔鬼们几乎花费了大部分的时间来腐化凡人（最常见的就是向凡人推销魔鬼契约），以此来扩大自 己在整个世界的影响力。在这方面，魔鬼们还是很守规矩的。&br&&b&a.魔鬼尊重契约和自由的意志&/b&&br&也就是说，他们不会强迫你签订契约，也不会违背契约，因为···那实在是太没品了·····&br&&b&b.&/b&&b&我们尊敬那些敢于付出代价改变命运的人！&/b&&br&&b&他们比那些整天只会祈祷，希望别人施舍的人更配得到幸福！”&/b&&br&——魔预者督尼（魔鬼）&br&在天界语里，命运和信仰是同一个词。&br&&b&而在魔鬼的词典里，命运和代价是同一个词。&/b&所以，在魔鬼们看来，凡有所求，必有代价。&br&&br&读到这，你是不是觉得：看样子，魔鬼还是蛮不错的嘛····&br&&br&对不起，魔鬼才没有那么好心呢。 魔鬼并不是“守信”，而是遵循交易原则，作为“交易”，必须是“钱货两讫”，否则就不能称之为“交易”，因为遵循了“交易原则”，在人的一方看起来，就有了“守信”的假象。&br&&br& 但事实上，作为交易的契约，其中必有各项条款。如果我们仔细去研究，就一定会发现，魔鬼却是发起交易并撰写契约的一方，魔鬼的契约内都含有某些隐藏附加条款；关键是，魔鬼会以不等价的事物谎称为等价事物进行交换。我们可以从一些西方文学中找到类似依据，比如《浮士德》中魔鬼与浮士德的交易，就是以人的灵魂作为交易品，与魔鬼交换物质世界的享乐，最终人的灵魂堕入地狱（参考马洛的《浮士德》）。&br&&br& 因此这里就带出一个问题——&br&究竟是人的灵魂贵重，还是这个物质世界的享受贵重？&br&&br& 作为交易的庄家，魔鬼很清楚人拥有什么事物，其中哪些最为昂贵，因此魔鬼会用迎合人私欲的事物作为筹码，诱使人以人自己视为不重要但对魔鬼来说非常有价值的事物作为交易品，在很多文学作品中，我们看到的都是魔鬼要求人用自己的灵魂、生命、良心等事物来交换，作为交易品，魔鬼所能给人的，就是物质世界的财富、权力、地位、美色、事业等事物，而这些也恰恰是人类在这个物质世界上所能看到的最有诱惑力的事物。并且人常常关注于眼睛所能看见的物质利益，对于眼睛看不见的比如灵魂、信仰、良心、来生等等，往往并不太关心。对于魔鬼来说，只要它得到了人的灵魂、生命、时间、良心、道德、信念等等，它就得到了自己想要的东西，而对于人在履行该交易中会经历什么、最后结局怎样，魔鬼既不负任何责任，也毫不关心，相反，它更乐意看见人因这个不公平条约而落入地狱的下场。&br&&br& 所以，某些以这些素材创作的西方文学作品，都会设定主角在面对魔鬼提出的交易时甘心接受魔鬼的交易条件而签下契约。&br&&br& 除了以浮士德的经历为题材的作品外，类似的例子还有以该依据为基础素材所创作的欧美漫画作品《幽灵骑士（Ghost Rider）》、梦工厂的动画长片《怪物史瑞克4》等。&br& 因此，魔鬼的“守信”只是一个假象，它给人开出了一个不公平的交易契约，以人眼所能看见的物质去换得人眼所看不见的生命、灵魂等事物。但人自身无法看透这个交易背后的不公平，就会错误地以为魔鬼“守信”。&br&&br&&br&&b&4.天堂与地狱之战&/b&&br&可参考&a href=&http://www.zhihu.com/question//answer/& class=&internal&&撒旦和耶和华之间有怎样的恩怨？战争结束了吗？ - 知乎用户的回答&/a&&br&
众神：他们为了巩固秩序体系创造了“信仰”，并且衍生出了“等级”这概念。&br&
地狱：我们创造了“思想”，同时衍生出了自由。&br&
众神：搞出了“天堂”，衍生出“救赎”。&br&
地狱：被迫成了“地狱”，但“选择”也随之衍生。随后我们用“怀疑”去稀释他们的“真理”；用“律法”去限制“正义”；用“代价”偷换了“奉献”；用“尊严”讽刺“荣誉”。&br&
这场战争打了很久，他们最后一个作品竟然叫“永恒”……然后，表层意义上的战争财正式展开。 &br&
“永恒”？对啊，有什么能对抗“不朽”呢？&br&
当听到“不朽”这个词语时，所有的魔鬼都笑了——于是“幽默”就此诞生。&br&
既然双方都必须接受同样的概念，那最后我们不都一样了吗？&br&
有本质的不同，我的主！&br&&b&
我们认为规则高于秩序，而他们认为秩序高于规则！&/b&&br&&br&------------------------------------------------------------------------------------------------&br&&b&恶魔与魔鬼的不同&/b&&br&&br&1.恶魔最重要的部位是心脏&br&
魔鬼最要的部位则是大脑&br&&br&&b&2.直面所有现实，这是我们魔鬼的本色！&/b&&br&&b&
只有那些神灵和恶魔才会觉得自己无所不能！&/b&&br&
——————第七君主莫达拉&br&&br&3.&b&恶魔的力量源于意志&/b&&br&&b&
魔鬼的力量则源于思想，&/b&通过思想见的逻辑包容性，我们会不断融合成为更严密的意志体。“融合”这个词或许可怕了点，用“认同”更合适一些。&br&&br&4.恶魔喜欢看到美的毁灭&br&
魔鬼则更喜欢看到美的堕落
根据不同的作品，对于恶魔或魔鬼的定义可能是不同的。下面是某一作品中对恶魔和魔鬼的描述： 恶魔：混乱邪恶的代表 居住在深渊，有许多种族，不同的深渊位面往往有不同的特有的恶魔，它们的身体结构截然不同，也有不同的能力，但是它们大多都有超越人类的肉…
一句话先回答问题：&b&因为斐波那契数列在数学和生活以及自然界中都非常有用。&/b&&br&&br&下面我就尽我所能，讲述一下斐波那契数列。&br&&br&&b&一、起源和定义&/b&&br&&br&斐波那契数列最早被提出是印度数学家Gopala，他在研究箱子包装物件长度恰好为1和2时的方法数时首先描述了这个数列。也就是这个问题：&br&&br&&blockquote&有n个台阶，你每次只能跨一阶或两阶，上楼有几种方法？&br&&/blockquote&&br&而最早研究这个数列的当然就是&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E6%E6%25B3%25A2%25E9%%25E5%25A5%2591& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契&/a&（Leonardo Fibonacci）了，他当时是为了描述如下情况的兔子生长数目：&br&&br&&blockquote&&ul&&li&第一个月初有一对刚诞生的兔子&/li&&li&第二个月之后（第三个月初）它们可以生育&/li&&li&每月每对可生育的兔子会诞生下一对新兔子&/li&&li&兔子永不死去&/li&&/ul&&/blockquote&&br&&figure&&img src=&https://pic3.zhimg.com/50/d4816eea15e9_b.jpg& data-rawwidth=&531& data-rawheight=&298& class=&origin_image zh-lightbox-thumb& width=&531& data-original=&https://pic3.zhimg.com/50/d4816eea15e9_r.jpg&&&/figure&&br&这个数列出自他赫赫有名的大作《计算之书》（没有维基词条，坑），后来就被广泛的应用于各种场合了。这个数列是这么定义的：&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/46c741e0cabc54bc_b.jpg& data-rawwidth=&469& data-rawheight=&122& class=&origin_image zh-lightbox-thumb& width=&469& data-original=&https://pic1.zhimg.com/50/46c741e0cabc54bc_r.jpg&&&/figure&&br&&a href=&//link.zhihu.com/?target=http%3A//oeis.org/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&The On-Line Encyclopedia of Integer Sequences(R) (OEIS(R))&/a&序号为&a href=&//link.zhihu.com/?target=http%3A//oeis.org/A000045& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&A000045 - OEIS&/a&&br&&br&（注意，并非满足第三条的都是斐波那契数列，&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E5%258D%25A2%25E5%258D%25A1%25E6%2596%25AF%25E6%%25E5%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&卢卡斯数列&/a&（&a href=&//link.zhihu.com/?target=http%3A//oeis.org/A000032& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&A000032 - OEIS&/a&）也满足这一特点，但初始项定义不同）&br&&br&&b&二、求解方法&/b&&br&&br&讲完了定义，再来说一说如何求对应的项。斐波那契数列是编程书中讲递归必提的，因为它是按照递归定义的。所以我们就从递归开始讲起。&br&&br&1.递归求解&br&&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&kt&&int&/span& &span class=&nf&&Fib&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&return&/span& &span class=&n&&n&/span& &span class=&o&&&&/span& &span class=&mi&&2&/span& &span class=&o&&?&/span& &span class=&mi&&1&/span& &span class=&o&&:&/span& &span class=&p&&(&/span&&span class=&n&&Fib&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span& &span class=&o&&+&/span& &span class=&n&&Fib&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&o&&-&/span&&span class=&mi&&2&/span&&span class=&p&&));&/span&
&span class=&p&&}&/span&
&/code&&/pre&&/div&&br&这是编程最方便的解法，当然，也是效率最低的解法，原因是会出现大量的重复计算。为了避免这种情况，可以采用递推的方式。&br&&br&2.递推求解&br&&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&kt&&int&/span& &span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&mi&&1000&/span&&span class=&p&&];&/span&
&span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&&span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&k&&for&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&i&/span& &span class=&o&&=&/span& &span class=&mi&&2&/span&&span class=&p&&;&/span&&span class=&n&&i&/span& &span class=&o&&&&/span& &span class=&mi&&1000&/span&&span class=&p&&;&/span&&span class=&n&&i&/span&&span class=&o&&++&/span&&span class=&p&&)&/span& &span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&+&/span& &span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&o&&-&/span&&span class=&mi&&2&/span&&span class=&p&&];&/span&
&/code&&/pre&&/div&&br&递推的方法可以在O(n)的时间内求出Fib(n)的值。但是这实际还是不够好，因为当n很大时这个算法还是无能为力的。接下来就要来讲一个有意思的东西：矩阵。&br&&br&3.矩阵递推关系&br&&br&学过代数的人可以看出，下面这个式子是成立的：&br&&img src=&//www.zhihu.com/equation?tex=%5Cbegin%7Bbmatrix%7D%0AFib%28n%2B1%29+%5C%5C%0AFib%28n%29%0A%5Cend+%7Bbmatrix%7D%0A%3D%5Cbegin%7Bbmatrix%7D%0A1%261+%5C%5C%0A1%260%0A%5Cend%7Bbmatrix%7D%0A%5Cbegin%7Bbmatrix%7D%0AFib%28n%29+%5C%5C%0AFib%28n-1%29%0A%5Cend%7Bbmatrix%7D& alt=&\begin{bmatrix}
Fib(n+1) \\
\end {bmatrix}
=\begin{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\end{bmatrix}& eeimg=&1&&&br&不停地利用这个式子迭代右边的列向量，会得到下面的式子：&br&&img src=&//www.zhihu.com/equation?tex=%5Cbegin%7Bbmatrix%7D%0AFib%28n%2B1%29+%5C%5C%0AFib%28n%29%0A%5Cend+%7Bbmatrix%7D%0A%3D%5Cbegin%7Bbmatrix%7D%0A1%261+%5C%5C%0A1%260%0A%5Cend%7Bbmatrix%7D%5E%7Bn%7D%0A%5Cbegin%7Bbmatrix%7D%0AFib%281%29+%5C%5C%0AFib%280%29%0A%5Cend%7Bbmatrix%7D& alt=&\begin{bmatrix}
Fib(n+1) \\
\end {bmatrix}
=\begin{bmatrix}
\end{bmatrix}^{n}
\begin{bmatrix}
\end{bmatrix}& eeimg=&1&&&br&这样，问题就转化为如何计算这个矩阵的n次方了，可以采用快速幂的方法。&a href=&//link.zhihu.com/?target=http%3A//baike.baidu.com/link%3Furl%3Ddbyx1Lo9P_Ca2cRdKuttAobLhFrmlyfGuCYn1MnzHvOVu-QwilkS3guqyxSZNXIWxlW8s9IIPAgf2_swv093Hq& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&快速幂_百度百科&/a&是利用结合律快速计算幂次的方法。比如我要计算&img src=&//www.zhihu.com/equation?tex=2%5E%7B20%7D+& alt=&2^{20} & eeimg=&1&&，我们知道&img src=&//www.zhihu.com/equation?tex=2%5E%7B20%7D+%3D++2%5E%7B16%7D+%2A+2%5E%7B4%7D++& alt=&2^{20} =
2^{16} * 2^{4}
& eeimg=&1&&，而&img src=&//www.zhihu.com/equation?tex=2%5E%7B2%7D+& alt=&2^{2} & eeimg=&1&&可以通过&img src=&//www.zhihu.com/equation?tex=2%5E%7B1%7D+%5Ctimes+2%5E%7B1%7D+& alt=&2^{1} \times 2^{1} & eeimg=&1&&来计算，&img src=&//www.zhihu.com/equation?tex=2%5E%7B4%7D+& alt=&2^{4} & eeimg=&1&&而可以通过&img src=&//www.zhihu.com/equation?tex=2%5E%7B2%7D%5Ctimes+2%5E%7B2%7D++& alt=&2^{2}\times 2^{2}
& eeimg=&1&&计算，以此类推。通过这种方法，可以在O（lbn）的时间里计算出一个数的n次幂。快速幂的代码如下：&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&kt&&int&/span& &span class=&nf&&Qpow&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&a&/span&&span class=&p&&,&/span&&span class=&kt&&int&/span& &span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&kt&&int&/span& &span class=&n&&ans&/span& &span class=&o&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&k&&while&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&if&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&o&&&&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span& &span class=&n&&ans&/span& &span class=&o&&*=&/span& &span class=&n&&a&/span&&span class=&p&&;&/span&
&span class=&n&&a&/span& &span class=&o&&*=&/span& &span class=&n&&a&/span&&span class=&p&&;&/span&
&span class=&n&&n&/span& &span class=&o&&&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&span class=&k&&return&/span& &span class=&n&&ans&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&/code&&/pre&&/div&将上述代码中的整型变量a变成矩阵，数的乘法变成矩阵乘法，就是矩阵快速幂了。比如用矩阵快速幂计算斐波那契数列：&br&&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&cp&&#include &cstdio&&/span&
&span class=&cp&&#include &iostream&&/span&
&span class=&n&&using&/span& &span class=&n&&namespace&/span& &span class=&n&&std&/span&&span class=&p&&;&/span&
&span class=&k&&const&/span& &span class=&kt&&int&/span& &span class=&n&&MOD&/span& &span class=&o&&=&/span& &span class=&mi&&10000&/span&&span class=&p&&;&/span&
&span class=&k&&struct&/span& &span class=&n&&matrix&/span&&span class=&c1&&//定义矩阵结构体&/span&
&span class=&p&&{&/span&
&span class=&kt&&int&/span& &span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&2&/span&&span class=&p&&][&/span&&span class=&mi&&2&/span&&span class=&p&&];&/span&
&span class=&p&&}&/span&&span class=&n&&ans&/span&&span class=&p&&,&/span& &span class=&n&&base&/span&&span class=&p&&;&/span&
&span class=&n&&matrix&/span& &span class=&nf&&multi&/span&&span class=&p&&(&/span&&span class=&n&&matrix&/span& &span class=&n&&a&/span&&span class=&p&&,&/span& &span class=&n&&matrix&/span& &span class=&n&&b&/span&&span class=&p&&)&/span&&span class=&c1&&//定义矩阵乘法&/span&
&span class=&p&&{&/span&
&span class=&n&&matrix&/span& &span class=&n&&tmp&/span&&span class=&p&&;&/span&
&span class=&k&&for&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&i&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span& &span class=&n&&i&/span& &span class=&o&&&&/span& &span class=&mi&&2&/span&&span class=&p&&;&/span& &span class=&o&&++&/span&&span class=&n&&i&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&for&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&j&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span& &span class=&n&&j&/span& &span class=&o&&&&/span& &span class=&mi&&2&/span&&span class=&p&&;&/span& &span class=&o&&++&/span&&span class=&n&&j&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&n&&tmp&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&][&/span&&span class=&n&&j&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&
&span class=&k&&for&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&k&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span& &span class=&n&&k&/span& &span class=&o&&&&/span& &span class=&mi&&2&/span&&span class=&p&&;&/span& &span class=&o&&++&/span&&span class=&n&&k&/span&&span class=&p&&)&/span&
&span class=&n&&tmp&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&][&/span&&span class=&n&&j&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&p&&(&/span&&span class=&n&&tmp&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&][&/span&&span class=&n&&j&/span&&span class=&p&&]&/span& &span class=&o&&+&/span& &span class=&n&&a&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&][&/span&&span class=&n&&k&/span&&span class=&p&&]&/span& &span class=&o&&*&/span& &span class=&n&&b&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&k&/span&&span class=&p&&][&/span&&span class=&n&&j&/span&&span class=&p&&])&/span& &span class=&o&&%&/span& &span class=&n&&MOD&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&span class=&p&&}&/span&
&span class=&k&&return&/span& &span class=&n&&tmp&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&span class=&kt&&int&/span& &span class=&nf&&fast_mod&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&c1&&// 求矩阵 base 的
n 次幂 &/span&
&span class=&p&&{&/span&
&span class=&n&&base&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&base&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&base&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&][&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&n&&base&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&
&span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&c1&&// ans 初始化为单位矩阵 &/span&
&span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&][&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&
&span class=&k&&while&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&if&/span&&span class=&p&&(&/span&&span class=&n&&n&/span& &span class=&o&&&&/span& &span class=&mi&&1&/span&&span class=&p&&)&/span&
&span class=&c1&&//实现 ans *= 其中要先把 ans赋值给 tmp，然后用 ans = tmp * t &/span&
&span class=&n&&ans&/span& &span class=&o&&=&/span& &span class=&n&&multi&/span&&span class=&p&&(&/span&&span class=&n&&ans&/span&&span class=&p&&,&/span& &span class=&n&&base&/span&&span class=&p&&);&/span&
&span class=&n&&base&/span& &span class=&o&&=&/span& &span class=&n&&multi&/span&&span class=&p&&(&/span&&span class=&n&&base&/span&&span class=&p&&,&/span& &span class=&n&&base&/span&&span class=&p&&);&/span&
&span class=&n&&n&/span& &span class=&o&&&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&span class=&k&&return&/span& &span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&];&/span&
&span class=&p&&}&/span&
&span class=&kt&&int&/span& &span class=&nf&&main&/span&&span class=&p&&()&/span&
&span class=&p&&{&/span&
&span class=&kt&&int&/span& &span class=&n&&n&/span&&span class=&p&&;&/span&
&span class=&k&&while&/span&&span class=&p&&(&/span&&span class=&n&&scanf&/span&&span class=&p&&(&/span&&span class=&s&&&%d&&/span&&span class=&p&&,&/span& &span class=&o&&&&/span&&span class=&n&&n&/span&&span class=&p&&)&/span& &span class=&o&&&&&/span& &span class=&n&&n&/span& &span class=&o&&!=&/span& &span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&n&&printf&/span&&span class=&p&&(&/span&&span class=&s&&&%d&/span&&span class=&se&&\n&/span&&span class=&s&&&&/span&&span class=&p&&,&/span& &span class=&n&&fast_mod&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&p&&));&/span&
&span class=&p&&}&/span&
&span class=&k&&return&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&/code&&/pre&&/div&&br&4.通项公式&br&&br&无论如何，对于一个数列，我们都是希望可以建立&img src=&//www.zhihu.com/equation?tex=F%28n%29& alt=&F(n)& eeimg=&1&&与n的关系，也就是通项公式，而用不同方法去求解通项公式也是很有意思的。&br&&br&（1）构造等比数列&br&&br&设&img src=&//www.zhihu.com/equation?tex=f%28n%29+%2B+%5Calpha+f%28n-1%29+%3D+%5Cbeta+%5Bf%28n-1%29+%2B+%5Calpha+f%28n-2%29%5D& alt=&f(n) + \alpha f(n-1) = \beta [f(n-1) + \alpha f(n-2)]& eeimg=&1&&,&br&化简得&img src=&//www.zhihu.com/equation?tex=f%28n%29%3D%28%5Cbeta+-%5Calpha+%29f%28n-1%29%2B%5Calpha+%5Cbeta+f%28n-2%29& alt=&f(n)=(\beta -\alpha )f(n-1)+\alpha \beta f(n-2)& eeimg=&1&&,&br&比较系数得&img src=&//www.zhihu.com/equation?tex=%5Cbeta+-%5Calpha+%3D1%2C%5Calpha+%5Cbeta+%3D1& alt=&\beta -\alpha =1,\alpha \beta =1& eeimg=&1&&，&br&解得&img src=&//www.zhihu.com/equation?tex=%5Calpha+%3D%5Cfrac%7B%5Csqrt%7B5%7D-1+%7D%7B2%7D+& alt=&\alpha =\frac{\sqrt{5}-1 }{2} & eeimg=&1&&，&img src=&//www.zhihu.com/equation?tex=%5Cbeta++%3D%5Cfrac%7B%5Csqrt%7B5%7D%2B1+%7D%7B2%7D+& alt=&\beta
=\frac{\sqrt{5}+1 }{2} & eeimg=&1&&&br&由于&img src=&//www.zhihu.com/equation?tex=f%28n%2B1%29%2B%5Calpha+f%28n%29%3D%5Bf%282%29%2B%5Calpha+f%281%29%5D%5Cbeta+%5E%7Bn-1%7D+%3D%5Cbeta+%5E%7Bn%7D+& alt=&f(n+1)+\alpha f(n)=[f(2)+\alpha f(1)]\beta ^{n-1} =\beta ^{n} & eeimg=&1&&&br&故有&img src=&//www.zhihu.com/equation?tex=%5Cfrac%7Bf%28n%2B1%29%7D%7B%5Cbeta+%5E%7Bn%2B1%7D%7D+%2B%5Cfrac%7B%5Calpha%7D%7B%5Cbeta+%7D+%5Cfrac%7Bf%28n%29%7D%7B%5Cbeta+%5E%7Bn%7D+%7D+%3D%5Cfrac%7B1%7D%7B%5Cbeta+%7D+& alt=&\frac{f(n+1)}{\beta ^{n+1}} +\frac{\alpha}{\beta } \frac{f(n)}{\beta ^{n} } =\frac{1}{\beta } & eeimg=&1&&，设&img src=&//www.zhihu.com/equation?tex=g%28n%29%3D%5Cfrac%7Bf%28n%29%7D%7B%5Cbeta+%5E%7Bn%7D%7D+& alt=&g(n)=\frac{f(n)}{\beta ^{n}} & eeimg=&1&&.则有&br&&img src=&//www.zhihu.com/equation?tex=g%28n%2B1%29%2B%5Cfrac%7B%5Calpha+%7D%7B%5Cbeta+%7D+g%28n%29%3D%5Cfrac%7B1%7D%7B%5Cbeta+%7D+& alt=&g(n+1)+\frac{\alpha }{\beta } g(n)=\frac{1}{\beta } & eeimg=&1&&,设&img src=&//www.zhihu.com/equation?tex=g%28n%2B1%29%2B%5Clambda+%3D-%5Cfrac%7B%5Calpha+%7D%7B%5Cbeta+%7D+%28g%28n%29%2B%5Clambda+%29& alt=&g(n+1)+\lambda =-\frac{\alpha }{\beta } (g(n)+\lambda )& eeimg=&1&&,&br&解得&img src=&//www.zhihu.com/equation?tex=%5Clambda+%3D-%5Cfrac%7B1%7D%7B%5Calpha+%2B%5Cbeta+%7D+& alt=&\lambda =-\frac{1}{\alpha +\beta } & eeimg=&1&&，即{&img src=&//www.zhihu.com/equation?tex=g%28n%29%2B%5Clambda+& alt=&g(n)+\lambda & eeimg=&1&&}是等比数列。这样就有&br&&img src=&//www.zhihu.com/equation?tex=g%28n%29%2B%5Clambda%3D%28-%5Cfrac%7B%5Calpha+%7D%7B%5Cbeta+%7D+%29%5E%7Bn-1%7D++%28%5Cfrac%7B1%7D%7B%5Cbeta+%7D+%2B%5Clambda+%29& alt=&g(n)+\lambda=(-\frac{\alpha }{\beta } )^{n-1}
(\frac{1}{\beta } +\lambda )& eeimg=&1&&&br&到了现在，把上述解出的结果全部带入上式，稍作变形，我们就可以写出斐波那契数列的通项公式了&br&&br&&br&&img src=&//www.zhihu.com/equation?tex=f%28n%29%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+%5B%28%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D+%29%5E%7Bn%7D-%28%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D+%29%5E%7Bn%7D%5D& alt=&f(n)=\frac{1}{\sqrt{5} } [(\frac{1+\sqrt{5} }{2} )^{n}-(\frac{1-\sqrt{5} }{2} )^{n}]& eeimg=&1&&&br&&br&&br&这个方法还是比较麻烦的，但是非常基础。事实上还有其他更简单的方法。&br&&br&（2）线性代数解法&br&这个解法首先用到&br&&img src=&//www.zhihu.com/equation?tex=%5Cbegin%7Bbmatrix%7D%0AFib%28n%2B1%29+%5C%5C%0AFib%28n%29%0A%5Cend+%7Bbmatrix%7D%0A%3D%5Cbegin%7Bbmatrix%7D%0A1%261+%5C%5C%0A1%260%0A%5Cend%7Bbmatrix%7D%5E%7Bn%7D%0A%5Cbegin%7Bbmatrix%7D%0AFib%281%29+%5C%5C%0AFib%280%29%0A%5Cend%7Bbmatrix%7D& alt=&\begin{bmatrix}
Fib(n+1) \\
\end {bmatrix}
=\begin{bmatrix}
\end{bmatrix}^{n}
\begin{bmatrix}
\end{bmatrix}& eeimg=&1&&&br&&br&公式，如果可以找到矩阵&img src=&//www.zhihu.com/equation?tex=P& alt=&P& eeimg=&1&&使得&img src=&//www.zhihu.com/equation?tex=PAP%5E%7B-1%7D& alt=&PAP^{-1}& eeimg=&1&&为对角阵，我们就可以求出通项。下面需要一些高等代数知识，没学过的可直接跳过。&br&首先令&img src=&//www.zhihu.com/equation?tex=%7C%5Clambda+E-A%7C%3D0& alt=&|\lambda E-A|=0& eeimg=&1&&,解得两个特征根&br&&img src=&//www.zhihu.com/equation?tex=%5Clambda_%7B1%7D%3D%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D+%2C%0A%5Clambda_%7B2%7D%3D%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D++& alt=&\lambda_{1}=\frac{1-\sqrt{5} }{2} ,
\lambda_{2}=\frac{1+\sqrt{5} }{2}
& eeimg=&1&&&br&两个特征向量为&br&&img src=&//www.zhihu.com/equation?tex=%5Calpha+_%7B1%7D%3D%5B1%2C%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D+%5D+%5E%7BT%7D%2C%5Calpha+_%7B2%7D%3D%5B1%2C%5Cfrac%7B-1%2B%5Csqrt%7B5%7D+%7D%7B2%7D+%5D+%5E%7BT%7D%2C& alt=&\alpha _{1}=[1,\frac{1+\sqrt{5} }{2} ] ^{T},\alpha _{2}=[1,\frac{-1+\sqrt{5} }{2} ] ^{T},& eeimg=&1&&则&br&&br&&img src=&//www.zhihu.com/equation?tex=P%3D%5Cbegin%7Bbmatrix%7D%0A1%261+%5C%5C%0A%5Cfrac%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D+%26+%5Cfrac%7B%5Csqrt%7B5%7D-1%7D%7B2%7D%0A%5Cend%7Bbmatrix%7D%2C%0AP%5E%7B-1%7D%3D%5Cbegin%7Bbmatrix%7D%0A%5Cfrac%7B1-%5Csqrt%7B5%7D%7D%7B2%7D%261+%5C%5C%0A%5Cfrac%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D%26-1%0A%5Cend%7Bbmatrix%7D& alt=&P=\begin{bmatrix}
\frac{1+\sqrt{5}}{2} & \frac{\sqrt{5}-1}{2}
\end{bmatrix},
P^{-1}=\begin{bmatrix}
\frac{1-\sqrt{5}}{2}&1 \\
\frac{1+\sqrt{5}}{2}&-1
\end{bmatrix}& eeimg=&1&&&br&而&br&&img src=&//www.zhihu.com/equation?tex=%28PAP%5E%7B-1%7D%29%5E%7Bn%7D%3DPAP%5E%7B-1%7DPAP%5E%7B-1%7D...PAP%5E%7B-1%7D%3DPA%28P%5E%7B-1%7DP%29A%28P%5E%7B-1%7DP%29A...AP%5E%7B-1%7D%3DPA%5E%7Bn%7DP%5E%7B-1%7D& alt=&(PAP^{-1})^{n}=PAP^{-1}PAP^{-1}...PAP^{-1}=PA(P^{-1}P)A(P^{-1}P)A...AP^{-1}=PA^{n}P^{-1}& eeimg=&1&&&br&解出&img src=&//www.zhihu.com/equation?tex=A%5E%7Bn%7D%3DP%5E%7B-1%7D%28PAP%5E%7B-1%7D%29%5E%7Bn%7DP& alt=&A^{n}=P^{-1}(PAP^{-1})^{n}P& eeimg=&1&&,中间矩阵的n次方可以直接求出来：&br&&br&&img src=&//www.zhihu.com/equation?tex=%5Cbegin%7Bbmatrix%7D%0A%5Cfrac+%7B1-%5Csqrt%7B5%7D%7D%7B2%7D%260+%5C%5C%0A0%26%5Cfrac+%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D%0A%5Cend%7Bbmatrix%7D%5E%7Bn%7D%3D%0A%5Cbegin%7Bbmatrix%7D%0A%28%5Cfrac+%7B1-%5Csqrt%7B5%7D%7D%7B2%7D%29%5E%7Bn%7D%260%5C%5C%0A0%26%28%5Cfrac+%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D%29%5En%0A%5Cend%7Bbmatrix%7D& alt=&\begin{bmatrix}
\frac {1-\sqrt{5}}{2}&0 \\
0&\frac {1+\sqrt{5}}{2}
\end{bmatrix}^{n}=
\begin{bmatrix}
(\frac {1-\sqrt{5}}{2})^{n}&0\\
0&(\frac {1+\sqrt{5}}{2})^n
\end{bmatrix}& eeimg=&1&&&br&&br&然后可以轻易得到通项公式，上边已经给出，这里不再赘述。&br&&br&（3）特征方程解法&br&&br&通过方法（2），我们可以推导出一般的线性递推数列的通项求解方法，也就是特征方程法。我们可以发现，对于这种数列，通项总是可以表示为&img src=&//www.zhihu.com/equation?tex=f%28n%29%3DC_%7B1%7D+%5Clambda+_%7B1%7D%5E%7Bn%7D+%2BC_%7B2%7D+%5Clambda+_%7B2%7D%5E%7Bn%7D& alt=&f(n)=C_{1} \lambda _{1}^{n} +C_{2} \lambda _{2}^{n}& eeimg=&1&&的形式，因此可以直接利用已知项求解&img src=&//www.zhihu.com/equation?tex=C_%7B1%7D& alt=&C_{1}& eeimg=&1&&，&img src=&//www.zhihu.com/equation?tex=C_%7B2%7D+& alt=&C_{2} & eeimg=&1&&。具体做法如下：&br&&br&a.由递推数列构造特征方程&img src=&//www.zhihu.com/equation?tex=x%5E%7B2%7D%3Dx%2B1& alt=&x^{2}=x+1& eeimg=&1&&，解出两个特征值&img src=&//www.zhihu.com/equation?tex=%5Clambda_%7B1%7D%3D%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D+%2C%0A%5Clambda_%7B2%7D%3D%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D++& alt=&\lambda_{1}=\frac{1-\sqrt{5} }{2} ,
\lambda_{2}=\frac{1+\sqrt{5} }{2}
& eeimg=&1&&。&br&&br&b.带入&img src=&//www.zhihu.com/equation?tex=f%280%29%2Cf%281%29& alt=&f(0),f(1)& eeimg=&1&&,列出如下方程：&br&&br&&img src=&//www.zhihu.com/equation?tex=C_%7B1%7D+%2BC_%7B1%7D+%3D0& alt=&C_{1} +C_{1} =0& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D+C_%7B1%7D+%2B%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D+C_%7B2%7D+%3D1& alt=&\frac{1-\sqrt{5} }{2} C_{1} +\frac{1+\sqrt{5} }{2} C_{2} =1& eeimg=&1&&&br&&br&解得&img src=&//www.zhihu.com/equation?tex=C_%7B1%7D+%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+%2CC_%7B2%7D+%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+.& alt=&C_{1} =-\frac{1}{\sqrt{5} } ,C_{2} =\frac{1}{\sqrt{5} } .& eeimg=&1&&这样直接写出通项公式，是比较简单的做法。&br&&br&（4）母函数法（此方法涉及组合数学知识）&br&&br&设斐波那契数列的母函数为&img src=&//www.zhihu.com/equation?tex=G%28x%29& alt=&G(x)& eeimg=&1&&,即&br&&img src=&//www.zhihu.com/equation?tex=G%28x%29%3DF_%7B0%7D+%2BF_%7B1%7Dx%2BF_%7B2%7Dx%5E%7B2%7D%2BL%2BF_%7Bn%7Dx%5E%7Bn%7D& alt=&G(x)=F_{0} +F_{1}x+F_{2}x^{2}+L+F_{n}x^{n}& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%3D1%2Bx%2B%28F_%7B0%7D%2BF_%7B1%7D%29x%5E%7B2%7D%2BL%2B%28F_%7Bn-1%7D+%2BF_%7Bn-2%7D%29x%5E%7Bn%7D& alt=&=1+x+(F_{0}+F_{1})x^{2}+L+(F_{n-1} +F_{n-2})x^{n}& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%3D1%2B%28x%2BF_%7B1%7Dx%5E%7B2%7D%2BF_%7B2%7Dx%5E%7B3%7D%2BL%2B%29%2B%28F_%7B0%7Dx%5E%7B2%7D%2BF_%7B1%7Dx%5E3%2BL%29& alt=&=1+(x+F_{1}x^{2}+F_{2}x^{3}+L+)+(F_{0}x^{2}+F_{1}x^3+L)& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%3D1%2B%28x%2Bx%5E%7B2%7D%29G%28x%29& alt=&=1+(x+x^{2})G(x)& eeimg=&1&&&br&解得&img src=&//www.zhihu.com/equation?tex=G%28x%29%3D%5Cfrac%7B1%7D%7B1-x-x%5E%7B2%7D%7D+%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D%5Cfrac%7B1%7D%7B1-%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7Dx%7D+-%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D%5Cfrac%7B1%7D%7B1-%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7Dx%7D& alt=&G(x)=\frac{1}{1-x-x^{2}} =\frac{1}{\sqrt{5} } \frac{1+\sqrt{5} }{2}\frac{1}{1-\frac{1+\sqrt{5} }{2}x} -\frac{1}{\sqrt{5} } \frac{1-\sqrt{5} }{2}\frac{1}{1-\frac{1-\sqrt{5} }{2}x}& eeimg=&1&&&br&&br&再由幂级数展开公式&img src=&//www.zhihu.com/equation?tex=%5Cfrac%7B1%7D%7B1-x%7D%3D1%2Bx%2Bx%5E%7B2%7D%2B& alt=&\frac{1}{1-x}=1+x+x^{2}+& eeimg=&1&&……&br&&br&合并同类项并与&img src=&//www.zhihu.com/equation?tex=G%28x%29& alt=&G(x)& eeimg=&1&&的系数比较即可。&br&&br&&br&&br&到这里，求解斐波那契数列通项的方法就说的差不多了。无论是计算机求解还是数学推导，都体现出了非常多的技巧。而斐波那契数列的许多特性，就更加有意思了。&br&&br&&b&三、斐波那契数列的数学性质&/b&&br&&br&1.与黄金比的关系&br&&br&由通项公式，求相邻两项的商的极限，结果是黄金比，所以斐波那契数列又称为黄金比数列。斐波那契数列和黄金比还和一个有趣的数学概念——连分数有关：&br&&figure&&img src=&https://pic1.zhimg.com/50/bccfc83e3913_b.jpg& data-rawwidth=&355& data-rawheight=&139& class=&content_image& width=&355&&&/figure&2.一些简单的规律&br&&br&（1）任意连续四个斐波那契数，可以构造出一个毕达哥拉斯三元组。如取1,1,2,3.&br&中间两数相乘再乘2 ==》 4&br&外层2数乘积==》3&br&中间两数平方和==》5&br&得到3,4,5.&br&下一组是5,12,13,，有兴趣的读者可以再试着推一推，证明也是容易的。&br&&br&（2）整除性&br&&br&&p&每3个连续的斐波那契数有且只有一个被2整除，&/p&&br&&p&每4个连续的斐波那契数有且只有一个被3整除，&/p&&br&&p&每5个连续的斐波那契数有且只有一个被5整除，&/p&&br&&p&每6个连续的斐波那契数有且只有一个被8整除，&/p&&br&&p&每7个连续的斐波那契数有且只有一个被13整除，&/p&&p&…………&/p&&p&每n个连续的斐波那契数有且只有一个被&img src=&//www.zhihu.com/equation?tex=f%28n%29& alt=&f(n)& eeimg=&1&&整除.&br&&/p&&br&&p&（3）一些恒等式&/p&&br&&figure&&img src=&https://pic2.zhimg.com/50/f004b9eada9b8101ffa3_b.jpg& data-rawwidth=&532& data-rawheight=&519& class=&origin_image zh-lightbox-thumb& width=&532& data-original=&https://pic2.zhimg.com/50/f004b9eada9b8101ffa3_r.jpg&&&/figure&&br&&br&&p&3.杨辉三角中的斐波那契数列&/p&&br&&p&如图所示，每条斜线上的数的和就构成斐波那契数列。&/p&&p&&figure&&img src=&https://pic4.zhimg.com/50/dd81f7f0ab92f_b.jpg& data-rawwidth=&690& data-rawheight=&381& class=&origin_image zh-lightbox-thumb& width=&690& data-original=&https://pic4.zhimg.com/50/dd81f7f0ab92f_r.jpg&&&/figure&&br&即有&img src=&//www.zhihu.com/equation?tex=f%28n%29%3DC_%7Bn-1%7D%5E%7B0%7D+%2BC_%7Bn-2%7D%5E%7B1%7D%2BL%2BC_%7Bn-1-m%7D%5E%7Bm%7D& alt=&f(n)=C_{n-1}^{0} +C_{n-2}^{1}+L+C_{n-1-m}^{m}& eeimg=&1&&&/p&&br&&p&4.相关数列：卢卡斯（Lucas）数列&/p&&br&&p&卢卡斯数列的定义除了第0项为2之外，与斐波那契数列完全一致。即&/p&&figure&&img src=&https://pic1.zhimg.com/50/7bba6df1b75c06ceae49_b.jpg& data-rawwidth=&447& data-rawheight=&93& class=&origin_image zh-lightbox-thumb& width=&447& data-original=&https://pic1.zhimg.com/50/7bba6df1b75c06ceae49_r.jpg&&&/figure&&br&&p&其通项公式为：&/p&&figure&&img src=&https://pic1.zhimg.com/50/29b3b6e5e4cb75f83ba8f0_b.jpg& data-rawwidth=&323& data-rawheight=&85& class=&content_image& width=&323&&&/figure&&br&&p&卢卡斯数列和斐波那契数列有这些关系：&/p&&br&&img src=&//www.zhihu.com/equation?tex=F_%7B2n%7D%3DF_%7Bn%7DL_%7Bn%7D+& alt=&F_{2n}=F_{n}L_{n} & eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=5F_%7Bn%7D%3DL_%7Bn-1%7D%2BL_%7Bn%2B1%7D& alt=&5F_{n}=L_{n-1}+L_{n+1}& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=L_%7Bn%7D%5E%7B2%7D%3D5+F_%7Bn%7D%5E%7B2%7D%2B4%28-1%29%5E%7Bn%7D& alt=&L_{n}^{2}=5 F_{n}^{2}+4(-1)^{n}& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%5Clim_%7Bn+%5Crightarrow+%5Cinfty+%7D%7B%5Cfrac%7BL_%7Bn%7D%7D%7BF_%7Bn%7D%7D+%7D+%3D%5Csqrt%7B5%7D+& alt=&\lim_{n \rightarrow \infty }{\frac{L_{n}}{F_{n}} } =\sqrt{5} & eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=L_%7Bn%7D%3DF_%7Bn-1%7D%2BF_%7Bn%2B1%7D& alt=&L_{n}=F_{n-1}+F_{n+1}& eeimg=&1&&&br&&br&&p&5.组合数学&/p&&br&&p&（1）一些恒等式&/p&&br&&figure&&img src=&https://pic3.zhimg.com/50/46aacf48efabbd3e9cb30_b.jpg& data-rawwidth=&482& data-rawheight=&481& class=&origin_image zh-lightbox-thumb& width=&482& data-original=&https://pic3.zhimg.com/50/46aacf48efabbd3e9cb30_r.jpg&&&/figure&&br&&p&（2）同余特性&/p&&br&&img src=&//www.zhihu.com/equation?tex=%28F%28m%29%2CF%28n%29%29%3D%28m%2Cn%29& alt=&(F(m),F(n))=(m,n)& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=F%28m%29%7CF%28n%29%5CLeftrightarrow+m%7Cn& alt=&F(m)|F(n)\Leftrightarrow m|n& eeimg=&1&&&br&&p&当p为大于5的素数时，有：&/p&&img src=&//www.zhihu.com/equation?tex=F%28p-%28%5Cfrac%7Bp%7D%7B5%7D%29%29%5Cequiv+0%28modp%29+& alt=&F(p-(\frac{p}{5}))\equiv 0(modp) & eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=F%28p%29%5Cequiv+%28%5Cfrac%7Bp%7D%7B5%7D%29+%28modp%29+& alt=&F(p)\equiv (\frac{p}{5}) (modp) & eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=F%28p%2B%28%5Cfrac%7Bp%7D%7B5%7D%29%29%5Cequiv+1%28modp%29+& alt=&F(p+(\frac{p}{5}))\equiv 1(modp) & eeimg=&1&&&br&&p&其中&/p&&figure&&img src=&https://pic2.zhimg.com/50/c8f7b8c87987_b.jpg& data-rawwidth=&275& data-rawheight=&61& class=&content_image& width=&275&&&/figure&&br&&p&斐波那契数列还有许许多多的性质，我就不再一一介绍了。跑题了这么久，终于开始要真正回答问题了：斐波那契数列有什么用？&/p&&br&&p&&b&四、斐波那契数列的应用&/b&&/p&&br&&p&1.算法&/p&&p&a.斐波那契堆&/p&&br&&blockquote&&p&斐波那契堆(Fibonacci heap)是计算机科学中最小堆有序树的集合。它和二项式堆有类似的性质，可用于实现合并优先队列。特点是不涉及删除元素的操作有O(1)的平摊时间，用途包括稠密图每次Decrease－key只要O(1)的平摊时间，和二项堆的O(lgn)相比是巨大的改进。&br&&/p&&br&&p&斐波那契堆由一组最小堆构成，这些最小堆是有根的无序树。可以进行插入、查找、合并和删除等操作&/p&&p&1）插入：创建一个仅包含一个节点的新的斐波纳契堆，然后执行堆合并&/p&&p&2）查找：由于用一个指针指向了具有最小值的根节点，因此查找最小的节点是平凡的操作。&/p&&p&3）合并：简单合并两个斐波纳契堆的根表。即把两个斐波纳契堆的所有树的根首尾衔接并置。&/p&&p&4）删除（释放）最小节点&/p&&p&分为三步：&/p&&ol&&li&查找最小的根节点并删除它，其所有的子节点都加入堆的根表，即它的子树都成为堆所包含的树；&/li&&li&需要查找并维护堆的最小根节点，但这耗时较大。为此，同时完成堆的维护：对堆当前包含的树的度数从低到高，迭代执行具有相同度数的树的合并并实现最小树化调整，使得堆包含的树具有不同的度数。这一步使用一个数组，数组下标为根节点的度数，数组的值为指向该根节点指针。如果发现具有相同度数的其他根节点则合并两棵树并维护该数组的状态。&/li&&li&对当前堆的所有根节点查找最小的根节点。&/li&&/ol&5）降低一个点的键值：对一个节点的键值降低后，自键值降低的节点开始自下而上的迭代执行下述操作，直至到根节点或一个未被标记（marked）节点为止：&br&&p&如果当前节点键值小于其父节点的键值，则把该节点及其子树摘下来作为堆的新树的根节点；其原父节点如果是被标记（marked）节点，则也被摘下来作为堆的新树的根节点；如果其原父节点不是被标记（marked）节点且不是根节点，则其原父节点被加标记。&/p&&p&如果堆的新树的根节点被标记（marked），则去除该标记。&/p&&p&6）删除节点：把被删除节点的键值调整为负无穷小，然后执行“降低一个节点的键值”算法，然后再执行“删除最小节点”算法。&/p&&p&&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E6%E6%25B3%25A2%25E9%%25E5%25A5%%25A0%2586& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契堆&/a&&br&&/p&&/blockquote&&br&b.欧几里得算法的时间复杂度&br&&br&欧几里得算法是求解两个正整数最大公约数的算法，又称辗转相除法。代码如下：&br&&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&kt&&int&/span& &span class=&nf&&gcd&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&a&/span&&span class=&p&&,&/span&&span class=&kt&&int&/span& &span class=&n&&b&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&return&/span& &span class=&n&&b&/span& &span class=&o&&?&/span& &span class=&n&&gcd&/span&&span class=&p&&(&/span&&span class=&n&&b&/span&&span class=&p&&,&/span&&span class=&n&&a&/span&&span class=&o&&%&/span&&span class=&n&&b&/span&&span class=&p&&)&/span& &span class=&o&&:&/span& &span class=&n&&a&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&/code&&/pre&&/div&&br&在最坏的情况下，我们可以证明，若a较小，需要计算的次数为n，则&img src=&//www.zhihu.com/equation?tex=a%3EF%28n-1%29& alt=&a&F(n-1)& eeimg=&1&&.虽然说一般分析的时候会当成对数阶，但数论最常用的欧几里得算法竟然与斐波那契数列有关，也确实是很让人吃惊呢。&br&&br&2.物理学：氢原子能级问题&br&&br&&blockquote&&p&假定我们现在有一些氢气原子，一个电子最初所处的位置是最低的能级（Ground lever of energy），属于稳定状态。它能获得一个能量子或二个能量子（Quanta of energy）而使它上升到第一能级或者第二能级。但是在第一级的电子如失掉一个能量子就会下降到最低能级，它如获得一个能量子就会上升到第二级来。&/p&&br&&p&现在研究气体吸收和放出能量的情形，假定最初电子是处在稳定状态即零能级，然后让它吸收能量，这电子可以跳到第1能级或第2能级。然后再让这气体放射能量，这时电子在1级能级的就要下降到0能级，而在第2能级的可能下降到0能级或者第1能级的位置去。&/p&&br&&p&电子所处的状态可能的情形是：1、2、3、5、8、13、21…种。这是斐波那契数列的一部份。&br&&/p&&/blockquote&&br&3.自然界：植物的生长&br&&br&科学家发现，一些植物的花瓣、萼片、果实的数目以及排列的方式上，都有一个神奇的规律，它们都非常符合著名的斐波那契数列。例如：蓟，它们的头部几乎呈球状。在下图中，你可以看到两条不同方向的螺旋。我们可以数一下，顺时针旋转的（和左边那条旋转方向相同）螺旋一共有13条，而逆时针旋转的则有21条。此外还有菊花、向日葵、松果、菠萝等都是按这种方式生长的。&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/bb88e0bfc466_b.jpg& data-rawwidth=&298& data-rawheight=&293& class=&content_image& width=&298&&&/figure&&br&&figure&&img src=&https://pic4.zhimg.com/50/1c0bf51aa055bd675d5c28_b.jpg& data-rawwidth=&374& data-rawheight=&327& class=&content_image& width=&374&&&/figure&&br&&br&还有菠萝、松子等，也都符合这个特点，一般会出现34,55，89和144这几个数字。&br&&br&&figure&&img src=&https://pic4.zhimg.com/50/dc8bdf9d021a9fdea46f1b_b.jpg& data-rawwidth=&229& data-rawheight=&198& class=&content_image& width=&229&&&/figure&&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/cab9_b.jpg& data-rawwidth=&193& data-rawheight=&191& class=&content_image& width=&193&&&/figure&&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/30de68d1de6d4b86dc52eb_b.jpg& data-rawwidth=&199& data-rawheight=&190& class=&content_image& width=&199&&&/figure&&br&&br&最后上一张“斐波那契树”的图片：&br&&figure&&img src=&https://pic4.zhimg.com/50/6c4a54fba4c66143abb03c_b.jpg& data-rawwidth=&250& data-rawheight=&247& class=&content_image& width=&250&&&/figure&是的，这玩意就长这样，这种植物是存在的。&br&&br&4.波浪理论与股市&br&&br&这个答主不懂，大家可自行阅读文章&a href=&//link.zhihu.com/?target=http%3A//www.sbo8.com/bolanglilun/38440.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&波浪理论斐波那契数列与黄金分割率&/a&。&br&不过波浪的形状确实符合下边要说的斐波那契螺旋：&br&&figure&&img src=&https://pic3.zhimg.com/50/6b173ce40bc99eba839c5b_b.jpg& data-rawwidth=&484& data-rawheight=&443& class=&origin_image zh-lightbox-thumb& width=&484& data-original=&https://pic3.zhimg.com/50/6b173ce40bc99eba839c5b_r.jpg&&&/figure&&br&&br&5.斐波那契螺旋&br&&br&斐波那契螺旋又称黄金螺旋，在自然界中广泛存在。如图是一个边长为斐波那契数列的正方形组成的矩形。&br&&figure&&img src=&https://pic3.zhimg.com/50/0db4b080e5b_b.jpg& data-rawwidth=&223& data-rawheight=&138& class=&content_image& width=&223&&&/figure&&br&（加一句：看着这个图，是不是能发现&figure&&img src=&https://pic4.zhimg.com/50/bbc05fe2da6e397b9cecd7_b.jpg& data-rawwidth=&337& data-rawheight=&39& class=&content_image& width=&337&&&/figure&是显而易见的？）&br&&br&这样连起来就是斐波那契螺旋了&br&&figure&&img src=&https://pic3.zhimg.com/50/e7f6e0bfdfee5c52dcf791_b.jpg& data-rawwidth=&223& data-rawheight=&138& class=&content_image& width=&223&&&/figure&&br&&br&贝壳螺旋轮廓线&br&&figure&&img src=&https://pic2.zhimg.com/50/977bdf99ace9c519b3da7d7_b.jpg& data-rawwidth=&963& data-rawheight=&365& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic2.zhimg.com/50/977bdf99ace9c519b3da7d7_r.jpg&&&/figure&&br&&br&&figure&&img src=&https://pic4.zhimg.com/50/be079bf49c6_b.jpg& data-rawwidth=&963& data-rawheight=&577& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic4.zhimg.com/50/be079bf49c6_r.jpg&&&/figure&&figure&&img src=&https://pic2.zhimg.com/50/3903ae1febca8eae63f038aacd631564_b.jpg& data-rawwidth=&963& data-rawheight=&723& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic2.zhimg.com/50/3903ae1febca8eae63f038aacd631564_r.jpg&&&/figure&&br&&br&&br&&figure&&img src=&https://pic3.zhimg.com/50/7312cfbc09ead2a2b570f063bd0c2af5_b.jpg& data-rawwidth=&963& data-rawheight=&723& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic3.zhimg.com/50/7312cfbc09ead2a2b570f063bd0c2af5_r.jpg&&&/figure&&br&&figure&&img src=&https://pic1.zhimg.com/50/0f99eeda9bf_b.jpg& data-rawwidth=&963& data-rawheight=&555& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic1.zhimg.com/50/0f99eeda9bf_r.jpg&&&/figure&&br&向日葵的生长&br&&figure&&img src=&https://pic3.zhimg.com/50/a40a8ec6d3bb9_b.jpg& data-rawwidth=&963& data-rawheight=&723& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic3.zhimg.com/50/a40a8ec6d3bb9_r.jpg&&&/figure&&br&神奇的花&br&&figure&&img src=&https://pic2.zhimg.com/50/db5d355d805ba7f96b576ac2a5ed1982_b.jpg& data-rawwidth=&723& data-rawheight=&723& class=&origin_image zh-lightbox-thumb& width=&723& data-original=&https://pic2.zhimg.com/50/db5d355d805ba7f96b576ac2a5ed1982_r.jpg&&&/figure&&br&&br&6.建筑学&br&&br&&figure&&img src=&https://pic4.zhimg.com/50/27d8c5e1cdf8e107a3752a_b.jpg& data-rawwidth=&600& data-rawheight=&400& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic4.zhimg.com/50/27d8c5e1cdf8e107a3752a_r.jpg&&&/figure&&br&&figure&&img src=&https://pic4.zhimg.com/50/8e5b641cadfdfe793c43a71edbbfd582_b.jpg& data-rawwidth=&490& data-rawheight=&367& class=&origin_image zh-lightbox-thumb& width=&490& data-original=&https://pic4.zhimg.com/50/8e5b641cadfdfe793c43a71edbbfd582_r.jpg&&&/figure&&br&&figure&&img src=&https://pic2.zhimg.com/50/e400a2c0dbfd90c_b.jpg& data-rawwidth=&580& data-rawheight=&359& class=&origin_image zh-lightbox-thumb& width=&580& data-original=&https://pic2.zhimg.com/50/e400a2c0dbfd90c_r.jpg&&&/figure&&br&&figure&&img src=&https://pic2.zhimg.com/50/6dc0cda9399d53ccaee6c1_b.jpg& data-rawwidth=&370& data-rawheight=&500& class=&content_image& width=&370&&&/figure&&br&7.据说一个小男孩参考斐波那契数列发明了太阳能电池树：&br&&br&&blockquote&一名13岁的男孩根据斐波那契数列&a href=&//link.zhihu.com/?target=http%3A//inhabitat.com/13-year-old-makes-solar-power-breakthrough-by-harnessing-the-fibonacci-sequence/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&发明了太阳能电池树&/a&，其产生的电力比太阳能光伏电池阵列多20-50%。&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E6%E6%25B3%25A2%25E9%%25E5%25A5%%%25E5%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列&/a&类似从0和1开始，之后的数是之前两数的和，如0，1，1，2，3，5，8，13，21...&a href=&//link.zhihu.com/?target=http%3A//www.amnh.org/nationalcenter/youngnaturalistawards/2011/aidan.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Aidan Dwye&/a&在观察树枝分叉时发现它的分布模式类似斐波那契数列，这是大自然演化的一种结果，可能有助于树叶进行光合作用。&br&因此，Dwye猜想为什么不按照斐波那契数列排列太阳能电池？他设计了太阳能电池树，发现它的输出电力提高了20%，每天接受光照的时间延长了2.5小时。&/blockquote&&br&8.斐波那契螺旋形的摇椅&br&&br&&blockquote&妈妈摇椅是设计师Patrick Messier为自己的妻子兼合作伙伴Sophie Fournier设计的，当时他们刚有了第一个宝宝。&br&&p&当Sophie宣布自己怀孕时，她说想要一把摇椅，但发现没有一把摇椅能满足美观舒适的标准，于是Patrick决定自己做一把。&/p&&p&于是就有了这把妈妈摇椅。像是一个飘在空中的丝带，由一片纤维玻璃做成，曲线服从&a href=&//link.zhihu.com/?target=http%3A//www.shejipi.com/2748.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列分布&/a&，经过特殊的高光聚氨酯处理。&/p&&/blockquote&&figure&&img src=&https://pic2.zhimg.com/50/7b83bf1c98af7dec0cf992_b.jpg& data-rawwidth=&600& data-rawheight=&388& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic2.zhimg.com/50/7b83bf1c98af7dec0cf992_r.jpg&&&/figure&&br&&br&&b&五、数学上的扩展&/b&&br&&br&（1）广义斐波那契数列&br&定义：&img src=&//www.zhihu.com/equation?tex=a%2Bb%2Cab%5Cin+Z& alt=&a+b,ab\in Z& eeimg=&1&&,数列&img src=&//www.zhihu.com/equation?tex=f%28n%29& alt=&f(n)& eeimg=&1&&满足：&br&&figure&&img src=&https://pic2.zhimg.com/50/aee0c8f3fb2be52c05fa1f8d4b2857d2_b.jpg& data-rawwidth=&326& data-rawheight=&95& class=&content_image& width=&326&&&/figure&其通项为：&br&&figure&&img src=&https://pic2.zhimg.com/50/452e6b08c4a063a6430b8_b.jpg& data-rawwidth=&117& data-rawheight=&57& class=&content_image& width=&117&&&/figure&当&img src=&//www.zhihu.com/equation?tex=a%2Bb%3D1%2Cab%3D-1& alt=&a+b=1,ab=-1& eeimg=&1&&时即为斐波那契数列。&br&&br&（2）反斐波那契数列&br&&br&定义：&img src=&//www.zhihu.com/equation?tex=g%28n%2B2%29%3Dg%28n%29-g%28n%2B1%29& alt=&g(n+2)=g(n)-g(n+1)& eeimg=&1&&&br&反斐波那契数列相邻项比值的极限为&img src=&//www.zhihu.com/equation?tex=-0.618& alt=&-0.618& eeimg=&1&&。&br&&br&（3）巴都万数列(&a href=&//link.zhihu.com/?target=http%3A//oeis.org/A000931& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&A000931 - OEIS&/a&)&br&斐波那契数列可以刻画矩形，而巴都万数列则刻画的是三角形。其定义如下：&br&&figure&&img src=&https://pic2.zhimg.com/50/dcdfbcaf5f4b_b.jpg& data-rawwidth=&211& data-rawheight=&68& class=&content_image& width=&211&&&/figure&&br&(4)未解之谜：角谷猜想&br&&br&对一个正整数，若为奇数则乘3加1，若为偶数则除以2，通过有限次这样的操作，能否使得该数变成1？&br&这个猜想和斐波那契数列又很大关系，具体的可以看&a href=&//link.zhihu.com/?target=http%3A//www.doc88.com/p-5.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&角谷猜想的斐波那契数列现象&/a&。&br&&br&&br&&b&六、总结&/b&&br&&br&斐波那契数列是各个学科中都出现的小滑头，它许多漂亮的性质让我们着迷。上文我所描述的这些只是它的冰山一角，权当抛砖引玉。大家读完了我的答案，还可以再结合自己的专业去看一些相关的资料，更好的去了解这个有趣的数列。&br&&br&&b&七、参考文献&/b&&br&&br&&br&&br&&br&[1]&a href=&//link.zhihu.com/?target=http%3A//www.hytc.cn/xsjl/szh/lec5.pdf& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://www.&/span&&span class=&visible&&hytc.cn/xsjl/szh/lec5.p&/span&&span class=&invisible&&df&/span&&span class=&ellipsis&&&/span&&/a&&br&[2]&a href=&//link.zhihu.com/?target=http%3A//blog.sina.com.cn/s/blog_77eb54a40100rpp9.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列_一米阳光&/a&&br&[3]&a href=&//link.zhihu.com/?target=http%3A//www.360doc.com/content/14/02.shtml& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列的规律性&/a&&br&[4]&a href=&//link.zhihu.com/?target=http%3A//www.cnbeta.com/articles/152349.htm& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&13岁男孩根据斐波那契数列发明太阳能电池树_cnBeta 人物_cnBeta.COM&/a&&br&[5]&a href=&//link.zhihu.com/?target=http%3A//www.shejipi.com/4547.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&服从斐波那契数列分布的妈妈摇椅&/a&&br&[6]&a href=&//link.zhihu.com/?target=http%3A//shiba.hpe.sh.cn/jiaoyanzu/wuli/ShowArticle.aspx%3FarticleId%3D990%26classId%3D5& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&从斐波那契数列到黄金分割&/a&&br&[7]&a href=&//link.zhihu.com/?target=http%3A//max.book118.com/html/82822.shtm& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契《计算之书》的研究.pdf 全文 文档投稿网&/a&&br&[8]&a href=&//link.zhihu.com/?target=http%3A//www.cnblogs.com/newpanderking/archive//2117328.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列&/a&&br&['9]&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/zh-tw/%25E6%E6%25B3%25A2%25E9%%25E5%25A5%%%25E5%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&費氏數列&/a&&br&[10]&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E5%25B7%25B4%25E9%2583%25BD%25E8%2590%25AC%25E6%%25E5%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&巴都萬數列&/a&&br&[11]&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E5%258D%25A2%25E5%258D%25A1%25E6%2596%25AF%25E6%& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&zh.wikipedia.org/wiki/%&/span&&span class=&invisible&&E5%8D%A2%E5%8D%A1%E6%96%AF%E6%95%B0&/span&&span class=&ellipsis&&&/span&&/a&&br&[12]&a href=&//link.zhihu.com/?target=http%3A//www.doc88.com/p-5.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&角谷猜想的斐波那契数列现象&/a&&br&[13]&a href=&//link.zhihu.com/?target=http%3A//oeis.org/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&The On-Line Encyclopedia of Integer Sequences(R) (OEIS(R))&/a&&br&[14]&a href=&//link.zhihu.com/?target=http%3A//www.sbo8.com/bolanglilun/38440.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&波浪理论斐波那契数列与黄金分割率&/a&
一句话先回答问题：因为斐波那契数列在数学和生活以及自然界中都非常有用。 下面我就尽我所能，讲述一下斐波那契数列。 一、起源和定义 斐波那契数列最早被提出是印度数学家Gopala，他在研究箱子包装物件长度恰好为1和2时的方法数时首先描述了这个数列。也…
刚才翻官网时找到官方答案了，顺便把其他人的也都放上来：&br&&a href=&//link.zhihu.com/?target=http%3A//piapro.net/intl/zh-cn_character.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&piapro.net&/a&&br&&br&MIKU &br&苍绿色 #39C5BB rgb(57, 197, 187) &figure&&img src=&https://pic1.zhimg.com/50/v2-b0dfda47f94fcb5234b5f_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&RIN &br&橙黄色 #FFA500 rgb(255, 165, 0)&br&&figure&&img src=&https://pic2.zhimg.com/50/v2-d1d85b885ea0fdb34f4bfde9_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&LEN &br&黄色 #FFE212 rgb(255, 226, 17)&figure&&img src=&https://pic4.zhimg.com/50/v2-9462eeebcace77d84ee60_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&LUKA &br&粉色 #FFBFCB rgb(255, 192, 203)&figure&&img src=&https://pic1.zhimg.com/50/v2-6c3acd9c65fc7c3c8bb0d_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&MEIKO &br&红色 #D80000 rgb(216, 0, 0)&figure&&img src=&https://pic4.zhimg.com/50/v2-844fbf323219ecfdd98bbec_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&KAITO &br&蓝色 #0000FF rgb(0, 0, 255)&figure&&img src=&https://pic3.zhimg.com/50/v2-6a1e411f3e1ff58e3d4388_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&
刚才翻官网时找到官方答案了，顺便把其他人的也都放上来：
MIKU 苍绿色 #39C5BB rgb(57, 197, 187) RIN 橙黄色 #FFA500 rgb(255, 165, 0) LEN 黄色 #FFE212 rgb(255, 226, 17) LUKA 粉色 #FFBFCB rgb(255, 192, 203) MEIKO 红色 #D80000 rgb(216…
金句王。肥皂终结者。&br&&br&&b&壹【金句王】 &/b&&br& (转载自豆瓣【整理+重新翻译和润色 by 慕容澈，注：破折号之后的是不同的译本。】 ）&br&&br&The only difference between a caprice and a life-long passion is that the caprice lasts a little longer. &br&逢场作戏和终身不渝之间的区别只在于逢场作戏稍微长一些。 &br&&br&When one is in love, one always begins by deceiveing one's self, and one always ends by deceiving others. That is what would calls a romance. &br&爱，始于自我欺骗，终于欺骗他人。这就是所谓的浪漫。 &br&——恋爱总是以自欺开始，以欺人结束。 &br&&br&The very essence of romance is uncertainty. &br&浪漫的精髓就在于它充满种种可能。 &br&&br&Man is a rational animal who always loses his temper when he is called upon to act in accordance with the dictates of reason. &br&人是理性动物，但当他被要求按照理性的要求行动时，可又要发脾气了。 &br&&br&My wallpaper and I are fighting a duel to the death. One or other of us has to go. &br&墙纸越来越破，而我越来越老，两者之间总有一个要先消失。——日，于左岸旅店，他的遗言。 &br&&br&No man is rich enough to buy back his own past. &br&-An Ideal Husband (1895) &br&没有人富有到可以赎回自己的过去。 &br&——改编自《理想的丈夫》 &br&&br&The truth is rarely pure and never simple. &br&-The Importance of Being Earnest (1895) &br&真相很少纯粹，也决不简单。 &br&——《不可儿嬉》 &br&&br&To love oneself is the beginning of a lifelong romance. &br&爱自己是终身浪漫的开始。 &br&——爱自己就是开始延续一生的罗曼史。 &br&&br&We are all in the gutter, but some of us are looking at the stars. &br&我们都生活在阴沟里，但仍有人仰望星空。 &br&&br&Most people discover when it is too late that the only things one never regrets are one's mistakes. &br&大多数人发现他们从未后悔的事情只是他们的错误，但发现时已经太晚了。 &br&&br&What is the chief cause of divorce? Marriage. &br&什么是离婚的主要原因？结婚。 &br&&br&When a love comes to an end, weaklings cry, efficient ones instantly find another love and wise already had one in reserve. &br&当爱情走到尽头，软弱者哭个不停，有效率的马上去寻找下一个目标，而聪明的早就预备了下一个。 &br&&br&No great artist ever sees things as they really are. If he did he would cease to be an artist.&br&伟大的艺术家所看到的，从来都不是世界的本来面目。一旦他看透了，他就不再是艺术家。&br&&br&I represent to you all the sins you have never had the courage to commit. &br&我给你们讲述的是所有你们没勇气去犯的罪孽。 &br&&br&One can always be kind to people one cares nothing about. &br&一个人总是可以善待他毫不在意的人。 &br&&br&We Irish are too
we are a nation of brilliant failures, but we are the greatest talkers since the Greeks. &br&我们爱尔兰人太诗意以至不能做诗人，我们的国家里充满才华横溢的失败者，可我们是自希腊人以来最伟大的说空话之人。 &br&&br&What seems to us as bitter trials are often blessings in disguise &br&看似痛苦的试炼的往往是伪装的祝福。 &br&&br&The advantage of the emotions is that they lead us astray. &br&情感的好处就是让我们误入歧途。 &br&&br&Over the piano was printed a notice: Please do not shoot the pianist. He is doing his best.&br&Personal Impressions of America (Leadville) (1883) &br&钢琴上贴着一条告示：请不要枪杀钢琴师，他已经尽力了。 &br&&br&The heart was made to be broken. &br&心是用来碎的。 &br&&br&The public is wonderfully tolerant. It forgives everything except genius. &br&公众惊人地宽容。他们可以原谅一切，除了天才。 &br&&br&Religions die when they are proved to be true. Science is the record of dead religions. &br&宗教一旦被证明是正确时就会消亡。科学便是已消亡宗教的记录。 &br&&br&Why was I born with such contemporaries &br&为什么我会和这样同时代的人一块出生呢？ &br&&br&A poet can survive everything but a misprint. &br&诗人可以从任何事件中存活，印刷错误除外。 &br&&br&Only the shallow know themselves. &br&-Phrases and Philosophies for the use of the Young (1894) &br&只有浅薄的人才了解自己。 &br&&br&The only way to get rid of temptation is to yield to it... I can resist everything but temptation. &br&摆脱诱惑的唯一方式是臣服于诱惑……我能抗拒一切，除了诱惑。 &br&&br&Discontent is the first step in the progress of a man or a nation &br&不满是个人或民族迈向进步的第一步。 &br&&br&I like to do all the talking myself. It saves time, and prevents arguments. &br&我喜欢自言自语，因为这样节约时间，而且不会有人跟我争论。 &br&&br&Quotation is a serviceable substitute for wit. &br&格言是智慧耐用的替代品。 &br&&br&A dreamer is one who can only find his way by moonlight, and his punishment is that he sees the dawn before the rest of the world. &br&梦想家只能在月光下找到前进的方向，他为此遭受的惩罚是比所有人提前看到曙光。 &br&&br&Every saint has a past and every sinner has a future. &br&每个圣人都有过去，每个罪人都有未来。 &br&&br&To live is the rarest thing in the world. Most people exist, that is all. &br&生活是世上最罕见的事情，大多数人只是存在，仅此而已。 &br&&br&I have nothing to declare except my genius. &br&除了我的天才，我没什么好申报的。 &br&&br&I like men who have a future and women who have a past &br&我喜欢有未来的男人和有过去的女人。 &br&&br&Pessimist: One who, when he has the choice of two evils, chooses both. &br&悲观主义者是这种人：当他可以从两种罪恶中选择时，他把两种都选了。 &br&&br&Society exists only in the real world there are only individuals. &br&社会仅仅以一种精神概念而存在，真实世界中只有个体存在。 &br&&br&What is a cynic? A man who knows the price of everything and the value of nothing &br&一个愤世嫉俗的人知道所有东西的价格，却不知道任何东西的价值。 &br&&br&I like persons better than principles, and I like persons with no principles better than anything else in the world. &br&我喜欢人甚于原则，此外我还喜欢没原则的人甚于世界上的一切。 &br&&br&It is the confession, not the priest, that gives us absolution &br&给我们赦免的，是忏悔而不是牧师。 &br&&br&I don't wa I want to live. &br&我不想谋生；我想生活。 &br&&br&Moderation is a fatal thing. Nothing succeeds like excess. &br&适度是极其致命的事情。过度带来的成功是无可比拟的。 &br&&br&The only thing worse than being talked about is not being talked about. &br&世上只有一件事比被人议论更糟糕，那就是没有人议论你。 &br&&br&When the gods wish to punish us, they answer our prayers &br&-An Ideal husband, 1893 &br&当神想惩罚我们时，他们就回应我们的祈祷。 &br&-《一个理想的丈夫》 &br&&br&&Life is never fair...And perhaps it is a good thing for most of us that it is not.& &br&-An Ideal Husband，1893 &br&生活从来不是公平的……而且，或许对我们大多数人来说，这是件好事。 &br&-《一个理想的丈夫》 &br&&br&How can a woman be expected to be happy with a man who insists on treating her as if she were a perfectly normal human being. &br&如果男人坚持把她当作一个完全正常的人，女人如何能期望会从他那里获得幸福。 &br&&br&All charming people, I fancy, are spoiled. It is the secret of their attraction &br&我想所有迷人的人都是被宠爱着的，这是他们吸引力来源的秘密。 &br&&br&Nothing is so aggravating than calmness. &br&没有比冷静更让人恼火的。 &br&&br&Popularity is the one insult I have never suffered. &br&声望是我从未经受的侮辱之一。 &br&&br&Ridicule is the tribute paid to the genius by the mediocrities. &br&奚落是庸才对天才的颂歌。 &br&&br&To do nothing at all is the most difficult thing in the world, the most difficult and the most intellectual &br&什么也不做是世上最难的事情，最困难并且最智慧。 &br&&br&A true friend stabs you in the front. &br&真朋友才会当面中伤你。 &br&&br&Ordinary riches can be stolen, real riches cannot. In your soul are infinitely precious things that cannot be taken from you. &br&平常的财宝会被偷走，而真正的财富则不会。你灵魂里无限珍贵的东西是无法被夺走的。 &br&&br&The well bred contradict other people. The wise contradict themselves. &br&教养良好的人处处和他人过不去，头脑聪明的人处处和自己过不去。 &br&&br&Hatred is blind, as well as love. &br&恨是盲目的，爱亦然。 &br&&br&Looking good and dressing well is a necessity. Having a purpose in life is not. &br&注意穿着打扮是必要的。而拥有生活目标却并非如此。 &br&&br&Always forgive your enemies - nothing annoys them so much. &br&永远宽恕你的敌人，没有什么能比这个更让他们恼怒的了。 &br&&br&Children begin by
as they grow o sometimes, they forgive them &br&孩子最初爱他们父母，等大一些他们评判父母；然后有些时候，他们原谅父母。 &br&&br&There are only two tragedies in life: one is not getting what one wants, and the other is getting it. &br&生活中只有两种悲剧:一个是没有得到我们想要的，另外一个是得到我们想要的。 &br&——世上只有两类悲剧，有些人总是不能遂愿，而有些人总是心想事成。 &br&&br&A gentleman is one who never hurts anyone's feelings unintentionally. &br&绅士就是从不无心伤害别人感觉的人。 &br&&br&Conversation about the weather is the last refuge of the unimaginative &br&谈论天气是无趣人类最后的避难所。 &br&&br&One's real life is often the life that one does not lead. &br&真实生活就通常就是我们无法掌控的生活。 &br&&br&Wickedness is a myth invented by good people to account for the curious attractiveness of others &br&邪恶是善良的人们编造的谎言，用来解释其他人的特殊魅力。 &br&&}
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求翻译：但你是一位非常善良的人，因为你会对我说，不要害怕你的纹身，你将继续教我学习英语。是什么意思？
但你是一位非常善良的人，因为你会对我说，不要害怕你的纹身，你将继续教我学习英语。
问题补充：
But you are a very kind person, because you will say to me, do not be afraid your tattoo, you will continue to teach me to learn English.
But you are a very good man, because you will say to me, don't be afraid of your tattoo, you will continue to teach me English.
But you are an extremely good person, because you can say to me, do not have to be afraid your model community, you will continue to teach me to study English.
But you are a very kind person, because you said to me, don't be afraid of your tattoo, you will continue to teach me English.
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请输入您需要翻译的文本！根据不同的作品，对于恶魔或魔鬼的定义可能是不同的。下面是某一作品中对恶魔和魔鬼的描述：&br&&br&&b&恶魔：混乱邪恶的代表&/b&&br&居住在深渊，有许多种族，不同的深渊位面往往有不同的特有的恶魔，它们的身体结构截然不同，也有不同的能力，但是它们大多都有超越人类的肉搏能力，并能使用各种类法术。&br&恶魔的信徒用死者的鲜血，作为对恶魔的献礼，鲜血让恶魔获得畏惧和服从。&br&&br&&b&1.邪恶&/b&&br&恶魔暴躁易怒、满怀恶意、独断暴力、毫无道德感而且无法预料，它们致力于破坏和毁灭一切事物（包括同类），并以此为毕生乐趣，它们经常入侵其他位面，带去毁灭与恐惧。&br&&br&&b&2.混乱&/b&&br&相对于守序的魔鬼，&b&住在无尽深渊中的恶魔们，则掌握了一切事物的本质——混乱。&/b&（熵增原理？）&br&&br&它们抗拒秩序，即使是深渊领主也无法让它们完全井井有条的做事。除非它们被法术控制，否则它们绝不可能团结，也不可能有计划的行动，上一刻的想法，下一刻就会被推翻。它们就是强大而邪恶的疯子。至于这帮家伙混乱到什么程度，看他们的老大、深渊意志的代表——深渊之主的表现就知道了：&br&&br&深渊之主， 一只无法用言语描述的存在，之所以无法描述，在于它太过混乱，混乱到连自身真名都没有，完全得不可名状……
如果说“地狱之主”代表着因为某种目的、某种**而犯下的罪恶，那祂就是毫无目的，单纯为了杀戮和毁灭而杀戮和毁灭。“深渊之主”是太古龙们为祂起得代号，便于称呼这样一位存在。&br&&br&“深渊之主”这位存在是彻头彻尾的疯子，谁也不知道祂下一步想做什么，因为连祂自己也不知道！它随时可能玩一把“自爆”，然后再花费漫长的时光从深渊里恢复——祂从来不会衡量得失之间的比例。 所以从来没有人去猜测它的想法，因为要是猜得中，恐怕那人也会怀疑自己已经被深渊气息侵染，开始丢掉引以为豪的智慧和冷静思考能力了，毕竟，“深渊之主”的世界，正常人永远不懂。&br&&br&老大都这德行，下面的就算脑袋稍微正常点，估计也好不到哪里去！（我一直深深地怀疑这帮脑残家伙为什么到现在还没被狡猾的魔鬼们搞死·····）&br&&br& ------------------------------------------------------------------------------------------------&br&&br&&b&魔鬼：守序邪恶的的代表，伪善的代名词&/b&&br&是区别与恶魔的另一种邪恶异界生物，这些守序而邪恶的生物居住于地狱之中，共有7位魔鬼统领，他们也被称为“地狱七君主”或“七撒旦（神之敌）”&br&&br&&b&1.伪善&/b&&br&什么是伪善？&br&“&b&伪善是邪恶向美德所表达的最崇高的敬意&/b&。”魔鬼如是说。&br& 残忍的恶魔？那只是凶兽罢了，&b&魔鬼&/b&&b&从来不屑于低级的杀戮，玩弄人心才是魔鬼的特长&/b&，否则的话，七宗罪里怎么没有“杀戮，残忍”这样的罪？因为那只是人性的阴暗面，不是魔鬼的赠礼。&br&&br&&b&2.守序&/b&&br&&b&地狱的法则是什么？守序的邪恶！&/b&&br&与混乱、善变的恶魔不同，魔鬼是来地狱的生物。目前魔鬼中数量最多的是巴特祖族，他们以强大的力量、邪恶的性格、无情但有效率的组织而恶名昭彰。巴特祖族有严谨的社会 街机系统，其中的全力不仅取决于力量，还取决于地位高低。&br&&br&&b&3.交易与契约&/b&&br&可参考&a href=&http://www.zhihu.com/question//answer/& class=&internal&&有人出卖灵魂给撒旦吗？过程是怎样的？ - 知乎用户的回答&/a&&br&魔鬼们几乎花费了大部分的时间来腐化凡人（最常见的就是向凡人推销魔鬼契约），以此来扩大自 己在整个世界的影响力。在这方面，魔鬼们还是很守规矩的。&br&&b&a.魔鬼尊重契约和自由的意志&/b&&br&也就是说，他们不会强迫你签订契约，也不会违背契约，因为···那实在是太没品了·····&br&&b&b.&/b&&b&我们尊敬那些敢于付出代价改变命运的人！&/b&&br&&b&他们比那些整天只会祈祷，希望别人施舍的人更配得到幸福！”&/b&&br&——魔预者督尼（魔鬼）&br&在天界语里，命运和信仰是同一个词。&br&&b&而在魔鬼的词典里，命运和代价是同一个词。&/b&所以，在魔鬼们看来，凡有所求，必有代价。&br&&br&读到这，你是不是觉得：看样子，魔鬼还是蛮不错的嘛····&br&&br&对不起，魔鬼才没有那么好心呢。 魔鬼并不是“守信”，而是遵循交易原则，作为“交易”，必须是“钱货两讫”，否则就不能称之为“交易”，因为遵循了“交易原则”，在人的一方看起来，就有了“守信”的假象。&br&&br& 但事实上，作为交易的契约，其中必有各项条款。如果我们仔细去研究，就一定会发现，魔鬼却是发起交易并撰写契约的一方，魔鬼的契约内都含有某些隐藏附加条款；关键是，魔鬼会以不等价的事物谎称为等价事物进行交换。我们可以从一些西方文学中找到类似依据，比如《浮士德》中魔鬼与浮士德的交易，就是以人的灵魂作为交易品，与魔鬼交换物质世界的享乐，最终人的灵魂堕入地狱（参考马洛的《浮士德》）。&br&&br& 因此这里就带出一个问题——&br&究竟是人的灵魂贵重，还是这个物质世界的享受贵重？&br&&br& 作为交易的庄家，魔鬼很清楚人拥有什么事物，其中哪些最为昂贵，因此魔鬼会用迎合人私欲的事物作为筹码，诱使人以人自己视为不重要但对魔鬼来说非常有价值的事物作为交易品，在很多文学作品中，我们看到的都是魔鬼要求人用自己的灵魂、生命、良心等事物来交换，作为交易品，魔鬼所能给人的，就是物质世界的财富、权力、地位、美色、事业等事物，而这些也恰恰是人类在这个物质世界上所能看到的最有诱惑力的事物。并且人常常关注于眼睛所能看见的物质利益，对于眼睛看不见的比如灵魂、信仰、良心、来生等等，往往并不太关心。对于魔鬼来说，只要它得到了人的灵魂、生命、时间、良心、道德、信念等等，它就得到了自己想要的东西，而对于人在履行该交易中会经历什么、最后结局怎样，魔鬼既不负任何责任，也毫不关心，相反，它更乐意看见人因这个不公平条约而落入地狱的下场。&br&&br& 所以，某些以这些素材创作的西方文学作品，都会设定主角在面对魔鬼提出的交易时甘心接受魔鬼的交易条件而签下契约。&br&&br& 除了以浮士德的经历为题材的作品外，类似的例子还有以该依据为基础素材所创作的欧美漫画作品《幽灵骑士（Ghost Rider）》、梦工厂的动画长片《怪物史瑞克4》等。&br& 因此，魔鬼的“守信”只是一个假象，它给人开出了一个不公平的交易契约，以人眼所能看见的物质去换得人眼所看不见的生命、灵魂等事物。但人自身无法看透这个交易背后的不公平，就会错误地以为魔鬼“守信”。&br&&br&&br&&b&4.天堂与地狱之战&/b&&br&可参考&a href=&http://www.zhihu.com/question//answer/& class=&internal&&撒旦和耶和华之间有怎样的恩怨？战争结束了吗？ - 知乎用户的回答&/a&&br&
众神：他们为了巩固秩序体系创造了“信仰”，并且衍生出了“等级”这概念。&br&
地狱：我们创造了“思想”，同时衍生出了自由。&br&
众神：搞出了“天堂”，衍生出“救赎”。&br&
地狱：被迫成了“地狱”，但“选择”也随之衍生。随后我们用“怀疑”去稀释他们的“真理”；用“律法”去限制“正义”；用“代价”偷换了“奉献”；用“尊严”讽刺“荣誉”。&br&
这场战争打了很久，他们最后一个作品竟然叫“永恒”……然后，表层意义上的战争财正式展开。 &br&
“永恒”？对啊，有什么能对抗“不朽”呢？&br&
当听到“不朽”这个词语时，所有的魔鬼都笑了——于是“幽默”就此诞生。&br&
既然双方都必须接受同样的概念，那最后我们不都一样了吗？&br&
有本质的不同，我的主！&br&&b&
我们认为规则高于秩序，而他们认为秩序高于规则！&/b&&br&&br&------------------------------------------------------------------------------------------------&br&&b&恶魔与魔鬼的不同&/b&&br&&br&1.恶魔最重要的部位是心脏&br&
魔鬼最要的部位则是大脑&br&&br&&b&2.直面所有现实，这是我们魔鬼的本色！&/b&&br&&b&
只有那些神灵和恶魔才会觉得自己无所不能！&/b&&br&
——————第七君主莫达拉&br&&br&3.&b&恶魔的力量源于意志&/b&&br&&b&
魔鬼的力量则源于思想，&/b&通过思想见的逻辑包容性，我们会不断融合成为更严密的意志体。“融合”这个词或许可怕了点，用“认同”更合适一些。&br&&br&4.恶魔喜欢看到美的毁灭&br&
魔鬼则更喜欢看到美的堕落
根据不同的作品，对于恶魔或魔鬼的定义可能是不同的。下面是某一作品中对恶魔和魔鬼的描述： 恶魔：混乱邪恶的代表 居住在深渊，有许多种族，不同的深渊位面往往有不同的特有的恶魔，它们的身体结构截然不同，也有不同的能力，但是它们大多都有超越人类的肉…
一句话先回答问题：&b&因为斐波那契数列在数学和生活以及自然界中都非常有用。&/b&&br&&br&下面我就尽我所能，讲述一下斐波那契数列。&br&&br&&b&一、起源和定义&/b&&br&&br&斐波那契数列最早被提出是印度数学家Gopala，他在研究箱子包装物件长度恰好为1和2时的方法数时首先描述了这个数列。也就是这个问题：&br&&br&&blockquote&有n个台阶，你每次只能跨一阶或两阶，上楼有几种方法？&br&&/blockquote&&br&而最早研究这个数列的当然就是&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E6%E6%25B3%25A2%25E9%%25E5%25A5%2591& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契&/a&（Leonardo Fibonacci）了，他当时是为了描述如下情况的兔子生长数目：&br&&br&&blockquote&&ul&&li&第一个月初有一对刚诞生的兔子&/li&&li&第二个月之后（第三个月初）它们可以生育&/li&&li&每月每对可生育的兔子会诞生下一对新兔子&/li&&li&兔子永不死去&/li&&/ul&&/blockquote&&br&&figure&&img src=&https://pic3.zhimg.com/50/d4816eea15e9_b.jpg& data-rawwidth=&531& data-rawheight=&298& class=&origin_image zh-lightbox-thumb& width=&531& data-original=&https://pic3.zhimg.com/50/d4816eea15e9_r.jpg&&&/figure&&br&这个数列出自他赫赫有名的大作《计算之书》（没有维基词条，坑），后来就被广泛的应用于各种场合了。这个数列是这么定义的：&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/46c741e0cabc54bc_b.jpg& data-rawwidth=&469& data-rawheight=&122& class=&origin_image zh-lightbox-thumb& width=&469& data-original=&https://pic1.zhimg.com/50/46c741e0cabc54bc_r.jpg&&&/figure&&br&&a href=&//link.zhihu.com/?target=http%3A//oeis.org/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&The On-Line Encyclopedia of Integer Sequences(R) (OEIS(R))&/a&序号为&a href=&//link.zhihu.com/?target=http%3A//oeis.org/A000045& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&A000045 - OEIS&/a&&br&&br&（注意，并非满足第三条的都是斐波那契数列，&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E5%258D%25A2%25E5%258D%25A1%25E6%2596%25AF%25E6%%25E5%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&卢卡斯数列&/a&（&a href=&//link.zhihu.com/?target=http%3A//oeis.org/A000032& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&A000032 - OEIS&/a&）也满足这一特点，但初始项定义不同）&br&&br&&b&二、求解方法&/b&&br&&br&讲完了定义，再来说一说如何求对应的项。斐波那契数列是编程书中讲递归必提的，因为它是按照递归定义的。所以我们就从递归开始讲起。&br&&br&1.递归求解&br&&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&kt&&int&/span& &span class=&nf&&Fib&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&return&/span& &span class=&n&&n&/span& &span class=&o&&&&/span& &span class=&mi&&2&/span& &span class=&o&&?&/span& &span class=&mi&&1&/span& &span class=&o&&:&/span& &span class=&p&&(&/span&&span class=&n&&Fib&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span& &span class=&o&&+&/span& &span class=&n&&Fib&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&o&&-&/span&&span class=&mi&&2&/span&&span class=&p&&));&/span&
&span class=&p&&}&/span&
&/code&&/pre&&/div&&br&这是编程最方便的解法，当然，也是效率最低的解法，原因是会出现大量的重复计算。为了避免这种情况，可以采用递推的方式。&br&&br&2.递推求解&br&&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&kt&&int&/span& &span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&mi&&1000&/span&&span class=&p&&];&/span&
&span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&&span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&k&&for&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&i&/span& &span class=&o&&=&/span& &span class=&mi&&2&/span&&span class=&p&&;&/span&&span class=&n&&i&/span& &span class=&o&&&&/span& &span class=&mi&&1000&/span&&span class=&p&&;&/span&&span class=&n&&i&/span&&span class=&o&&++&/span&&span class=&p&&)&/span& &span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&+&/span& &span class=&n&&Fib&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&o&&-&/span&&span class=&mi&&2&/span&&span class=&p&&];&/span&
&/code&&/pre&&/div&&br&递推的方法可以在O(n)的时间内求出Fib(n)的值。但是这实际还是不够好，因为当n很大时这个算法还是无能为力的。接下来就要来讲一个有意思的东西：矩阵。&br&&br&3.矩阵递推关系&br&&br&学过代数的人可以看出，下面这个式子是成立的：&br&&img src=&//www.zhihu.com/equation?tex=%5Cbegin%7Bbmatrix%7D%0AFib%28n%2B1%29+%5C%5C%0AFib%28n%29%0A%5Cend+%7Bbmatrix%7D%0A%3D%5Cbegin%7Bbmatrix%7D%0A1%261+%5C%5C%0A1%260%0A%5Cend%7Bbmatrix%7D%0A%5Cbegin%7Bbmatrix%7D%0AFib%28n%29+%5C%5C%0AFib%28n-1%29%0A%5Cend%7Bbmatrix%7D& alt=&\begin{bmatrix}
Fib(n+1) \\
\end {bmatrix}
=\begin{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\end{bmatrix}& eeimg=&1&&&br&不停地利用这个式子迭代右边的列向量，会得到下面的式子：&br&&img src=&//www.zhihu.com/equation?tex=%5Cbegin%7Bbmatrix%7D%0AFib%28n%2B1%29+%5C%5C%0AFib%28n%29%0A%5Cend+%7Bbmatrix%7D%0A%3D%5Cbegin%7Bbmatrix%7D%0A1%261+%5C%5C%0A1%260%0A%5Cend%7Bbmatrix%7D%5E%7Bn%7D%0A%5Cbegin%7Bbmatrix%7D%0AFib%281%29+%5C%5C%0AFib%280%29%0A%5Cend%7Bbmatrix%7D& alt=&\begin{bmatrix}
Fib(n+1) \\
\end {bmatrix}
=\begin{bmatrix}
\end{bmatrix}^{n}
\begin{bmatrix}
\end{bmatrix}& eeimg=&1&&&br&这样，问题就转化为如何计算这个矩阵的n次方了，可以采用快速幂的方法。&a href=&//link.zhihu.com/?target=http%3A//baike.baidu.com/link%3Furl%3Ddbyx1Lo9P_Ca2cRdKuttAobLhFrmlyfGuCYn1MnzHvOVu-QwilkS3guqyxSZNXIWxlW8s9IIPAgf2_swv093Hq& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&快速幂_百度百科&/a&是利用结合律快速计算幂次的方法。比如我要计算&img src=&//www.zhihu.com/equation?tex=2%5E%7B20%7D+& alt=&2^{20} & eeimg=&1&&，我们知道&img src=&//www.zhihu.com/equation?tex=2%5E%7B20%7D+%3D++2%5E%7B16%7D+%2A+2%5E%7B4%7D++& alt=&2^{20} =
2^{16} * 2^{4}
& eeimg=&1&&，而&img src=&//www.zhihu.com/equation?tex=2%5E%7B2%7D+& alt=&2^{2} & eeimg=&1&&可以通过&img src=&//www.zhihu.com/equation?tex=2%5E%7B1%7D+%5Ctimes+2%5E%7B1%7D+& alt=&2^{1} \times 2^{1} & eeimg=&1&&来计算，&img src=&//www.zhihu.com/equation?tex=2%5E%7B4%7D+& alt=&2^{4} & eeimg=&1&&而可以通过&img src=&//www.zhihu.com/equation?tex=2%5E%7B2%7D%5Ctimes+2%5E%7B2%7D++& alt=&2^{2}\times 2^{2}
& eeimg=&1&&计算，以此类推。通过这种方法，可以在O（lbn）的时间里计算出一个数的n次幂。快速幂的代码如下：&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&kt&&int&/span& &span class=&nf&&Qpow&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&a&/span&&span class=&p&&,&/span&&span class=&kt&&int&/span& &span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&kt&&int&/span& &span class=&n&&ans&/span& &span class=&o&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&k&&while&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&if&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&o&&&&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span& &span class=&n&&ans&/span& &span class=&o&&*=&/span& &span class=&n&&a&/span&&span class=&p&&;&/span&
&span class=&n&&a&/span& &span class=&o&&*=&/span& &span class=&n&&a&/span&&span class=&p&&;&/span&
&span class=&n&&n&/span& &span class=&o&&&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&span class=&k&&return&/span& &span class=&n&&ans&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&/code&&/pre&&/div&将上述代码中的整型变量a变成矩阵，数的乘法变成矩阵乘法，就是矩阵快速幂了。比如用矩阵快速幂计算斐波那契数列：&br&&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&cp&&#include &cstdio&&/span&
&span class=&cp&&#include &iostream&&/span&
&span class=&n&&using&/span& &span class=&n&&namespace&/span& &span class=&n&&std&/span&&span class=&p&&;&/span&
&span class=&k&&const&/span& &span class=&kt&&int&/span& &span class=&n&&MOD&/span& &span class=&o&&=&/span& &span class=&mi&&10000&/span&&span class=&p&&;&/span&
&span class=&k&&struct&/span& &span class=&n&&matrix&/span&&span class=&c1&&//定义矩阵结构体&/span&
&span class=&p&&{&/span&
&span class=&kt&&int&/span& &span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&2&/span&&span class=&p&&][&/span&&span class=&mi&&2&/span&&span class=&p&&];&/span&
&span class=&p&&}&/span&&span class=&n&&ans&/span&&span class=&p&&,&/span& &span class=&n&&base&/span&&span class=&p&&;&/span&
&span class=&n&&matrix&/span& &span class=&nf&&multi&/span&&span class=&p&&(&/span&&span class=&n&&matrix&/span& &span class=&n&&a&/span&&span class=&p&&,&/span& &span class=&n&&matrix&/span& &span class=&n&&b&/span&&span class=&p&&)&/span&&span class=&c1&&//定义矩阵乘法&/span&
&span class=&p&&{&/span&
&span class=&n&&matrix&/span& &span class=&n&&tmp&/span&&span class=&p&&;&/span&
&span class=&k&&for&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&i&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span& &span class=&n&&i&/span& &span class=&o&&&&/span& &span class=&mi&&2&/span&&span class=&p&&;&/span& &span class=&o&&++&/span&&span class=&n&&i&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&for&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&j&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span& &span class=&n&&j&/span& &span class=&o&&&&/span& &span class=&mi&&2&/span&&span class=&p&&;&/span& &span class=&o&&++&/span&&span class=&n&&j&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&n&&tmp&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&][&/span&&span class=&n&&j&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&
&span class=&k&&for&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&k&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span& &span class=&n&&k&/span& &span class=&o&&&&/span& &span class=&mi&&2&/span&&span class=&p&&;&/span& &span class=&o&&++&/span&&span class=&n&&k&/span&&span class=&p&&)&/span&
&span class=&n&&tmp&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&][&/span&&span class=&n&&j&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&p&&(&/span&&span class=&n&&tmp&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&][&/span&&span class=&n&&j&/span&&span class=&p&&]&/span& &span class=&o&&+&/span& &span class=&n&&a&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&][&/span&&span class=&n&&k&/span&&span class=&p&&]&/span& &span class=&o&&*&/span& &span class=&n&&b&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&n&&k&/span&&span class=&p&&][&/span&&span class=&n&&j&/span&&span class=&p&&])&/span& &span class=&o&&%&/span& &span class=&n&&MOD&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&span class=&p&&}&/span&
&span class=&k&&return&/span& &span class=&n&&tmp&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&span class=&kt&&int&/span& &span class=&nf&&fast_mod&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&c1&&// 求矩阵 base 的
n 次幂 &/span&
&span class=&p&&{&/span&
&span class=&n&&base&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&base&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&base&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&][&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&n&&base&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&
&span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&c1&&// ans 初始化为单位矩阵 &/span&
&span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&][&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&
&span class=&k&&while&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&if&/span&&span class=&p&&(&/span&&span class=&n&&n&/span& &span class=&o&&&&/span& &span class=&mi&&1&/span&&span class=&p&&)&/span&
&span class=&c1&&//实现 ans *= 其中要先把 ans赋值给 tmp，然后用 ans = tmp * t &/span&
&span class=&n&&ans&/span& &span class=&o&&=&/span& &span class=&n&&multi&/span&&span class=&p&&(&/span&&span class=&n&&ans&/span&&span class=&p&&,&/span& &span class=&n&&base&/span&&span class=&p&&);&/span&
&span class=&n&&base&/span& &span class=&o&&=&/span& &span class=&n&&multi&/span&&span class=&p&&(&/span&&span class=&n&&base&/span&&span class=&p&&,&/span& &span class=&n&&base&/span&&span class=&p&&);&/span&
&span class=&n&&n&/span& &span class=&o&&&&=&/span& &span class=&mi&&1&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&span class=&k&&return&/span& &span class=&n&&ans&/span&&span class=&p&&.&/span&&span class=&n&&m&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&][&/span&&span class=&mi&&1&/span&&span class=&p&&];&/span&
&span class=&p&&}&/span&
&span class=&kt&&int&/span& &span class=&nf&&main&/span&&span class=&p&&()&/span&
&span class=&p&&{&/span&
&span class=&kt&&int&/span& &span class=&n&&n&/span&&span class=&p&&;&/span&
&span class=&k&&while&/span&&span class=&p&&(&/span&&span class=&n&&scanf&/span&&span class=&p&&(&/span&&span class=&s&&&%d&&/span&&span class=&p&&,&/span& &span class=&o&&&&/span&&span class=&n&&n&/span&&span class=&p&&)&/span& &span class=&o&&&&&/span& &span class=&n&&n&/span& &span class=&o&&!=&/span& &span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&n&&printf&/span&&span class=&p&&(&/span&&span class=&s&&&%d&/span&&span class=&se&&\n&/span&&span class=&s&&&&/span&&span class=&p&&,&/span& &span class=&n&&fast_mod&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&p&&));&/span&
&span class=&p&&}&/span&
&span class=&k&&return&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&/code&&/pre&&/div&&br&4.通项公式&br&&br&无论如何，对于一个数列，我们都是希望可以建立&img src=&//www.zhihu.com/equation?tex=F%28n%29& alt=&F(n)& eeimg=&1&&与n的关系，也就是通项公式，而用不同方法去求解通项公式也是很有意思的。&br&&br&（1）构造等比数列&br&&br&设&img src=&//www.zhihu.com/equation?tex=f%28n%29+%2B+%5Calpha+f%28n-1%29+%3D+%5Cbeta+%5Bf%28n-1%29+%2B+%5Calpha+f%28n-2%29%5D& alt=&f(n) + \alpha f(n-1) = \beta [f(n-1) + \alpha f(n-2)]& eeimg=&1&&,&br&化简得&img src=&//www.zhihu.com/equation?tex=f%28n%29%3D%28%5Cbeta+-%5Calpha+%29f%28n-1%29%2B%5Calpha+%5Cbeta+f%28n-2%29& alt=&f(n)=(\beta -\alpha )f(n-1)+\alpha \beta f(n-2)& eeimg=&1&&,&br&比较系数得&img src=&//www.zhihu.com/equation?tex=%5Cbeta+-%5Calpha+%3D1%2C%5Calpha+%5Cbeta+%3D1& alt=&\beta -\alpha =1,\alpha \beta =1& eeimg=&1&&，&br&解得&img src=&//www.zhihu.com/equation?tex=%5Calpha+%3D%5Cfrac%7B%5Csqrt%7B5%7D-1+%7D%7B2%7D+& alt=&\alpha =\frac{\sqrt{5}-1 }{2} & eeimg=&1&&，&img src=&//www.zhihu.com/equation?tex=%5Cbeta++%3D%5Cfrac%7B%5Csqrt%7B5%7D%2B1+%7D%7B2%7D+& alt=&\beta
=\frac{\sqrt{5}+1 }{2} & eeimg=&1&&&br&由于&img src=&//www.zhihu.com/equation?tex=f%28n%2B1%29%2B%5Calpha+f%28n%29%3D%5Bf%282%29%2B%5Calpha+f%281%29%5D%5Cbeta+%5E%7Bn-1%7D+%3D%5Cbeta+%5E%7Bn%7D+& alt=&f(n+1)+\alpha f(n)=[f(2)+\alpha f(1)]\beta ^{n-1} =\beta ^{n} & eeimg=&1&&&br&故有&img src=&//www.zhihu.com/equation?tex=%5Cfrac%7Bf%28n%2B1%29%7D%7B%5Cbeta+%5E%7Bn%2B1%7D%7D+%2B%5Cfrac%7B%5Calpha%7D%7B%5Cbeta+%7D+%5Cfrac%7Bf%28n%29%7D%7B%5Cbeta+%5E%7Bn%7D+%7D+%3D%5Cfrac%7B1%7D%7B%5Cbeta+%7D+& alt=&\frac{f(n+1)}{\beta ^{n+1}} +\frac{\alpha}{\beta } \frac{f(n)}{\beta ^{n} } =\frac{1}{\beta } & eeimg=&1&&，设&img src=&//www.zhihu.com/equation?tex=g%28n%29%3D%5Cfrac%7Bf%28n%29%7D%7B%5Cbeta+%5E%7Bn%7D%7D+& alt=&g(n)=\frac{f(n)}{\beta ^{n}} & eeimg=&1&&.则有&br&&img src=&//www.zhihu.com/equation?tex=g%28n%2B1%29%2B%5Cfrac%7B%5Calpha+%7D%7B%5Cbeta+%7D+g%28n%29%3D%5Cfrac%7B1%7D%7B%5Cbeta+%7D+& alt=&g(n+1)+\frac{\alpha }{\beta } g(n)=\frac{1}{\beta } & eeimg=&1&&,设&img src=&//www.zhihu.com/equation?tex=g%28n%2B1%29%2B%5Clambda+%3D-%5Cfrac%7B%5Calpha+%7D%7B%5Cbeta+%7D+%28g%28n%29%2B%5Clambda+%29& alt=&g(n+1)+\lambda =-\frac{\alpha }{\beta } (g(n)+\lambda )& eeimg=&1&&,&br&解得&img src=&//www.zhihu.com/equation?tex=%5Clambda+%3D-%5Cfrac%7B1%7D%7B%5Calpha+%2B%5Cbeta+%7D+& alt=&\lambda =-\frac{1}{\alpha +\beta } & eeimg=&1&&，即{&img src=&//www.zhihu.com/equation?tex=g%28n%29%2B%5Clambda+& alt=&g(n)+\lambda & eeimg=&1&&}是等比数列。这样就有&br&&img src=&//www.zhihu.com/equation?tex=g%28n%29%2B%5Clambda%3D%28-%5Cfrac%7B%5Calpha+%7D%7B%5Cbeta+%7D+%29%5E%7Bn-1%7D++%28%5Cfrac%7B1%7D%7B%5Cbeta+%7D+%2B%5Clambda+%29& alt=&g(n)+\lambda=(-\frac{\alpha }{\beta } )^{n-1}
(\frac{1}{\beta } +\lambda )& eeimg=&1&&&br&到了现在，把上述解出的结果全部带入上式，稍作变形，我们就可以写出斐波那契数列的通项公式了&br&&br&&br&&img src=&//www.zhihu.com/equation?tex=f%28n%29%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+%5B%28%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D+%29%5E%7Bn%7D-%28%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D+%29%5E%7Bn%7D%5D& alt=&f(n)=\frac{1}{\sqrt{5} } [(\frac{1+\sqrt{5} }{2} )^{n}-(\frac{1-\sqrt{5} }{2} )^{n}]& eeimg=&1&&&br&&br&&br&这个方法还是比较麻烦的，但是非常基础。事实上还有其他更简单的方法。&br&&br&（2）线性代数解法&br&这个解法首先用到&br&&img src=&//www.zhihu.com/equation?tex=%5Cbegin%7Bbmatrix%7D%0AFib%28n%2B1%29+%5C%5C%0AFib%28n%29%0A%5Cend+%7Bbmatrix%7D%0A%3D%5Cbegin%7Bbmatrix%7D%0A1%261+%5C%5C%0A1%260%0A%5Cend%7Bbmatrix%7D%5E%7Bn%7D%0A%5Cbegin%7Bbmatrix%7D%0AFib%281%29+%5C%5C%0AFib%280%29%0A%5Cend%7Bbmatrix%7D& alt=&\begin{bmatrix}
Fib(n+1) \\
\end {bmatrix}
=\begin{bmatrix}
\end{bmatrix}^{n}
\begin{bmatrix}
\end{bmatrix}& eeimg=&1&&&br&&br&公式，如果可以找到矩阵&img src=&//www.zhihu.com/equation?tex=P& alt=&P& eeimg=&1&&使得&img src=&//www.zhihu.com/equation?tex=PAP%5E%7B-1%7D& alt=&PAP^{-1}& eeimg=&1&&为对角阵，我们就可以求出通项。下面需要一些高等代数知识，没学过的可直接跳过。&br&首先令&img src=&//www.zhihu.com/equation?tex=%7C%5Clambda+E-A%7C%3D0& alt=&|\lambda E-A|=0& eeimg=&1&&,解得两个特征根&br&&img src=&//www.zhihu.com/equation?tex=%5Clambda_%7B1%7D%3D%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D+%2C%0A%5Clambda_%7B2%7D%3D%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D++& alt=&\lambda_{1}=\frac{1-\sqrt{5} }{2} ,
\lambda_{2}=\frac{1+\sqrt{5} }{2}
& eeimg=&1&&&br&两个特征向量为&br&&img src=&//www.zhihu.com/equation?tex=%5Calpha+_%7B1%7D%3D%5B1%2C%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D+%5D+%5E%7BT%7D%2C%5Calpha+_%7B2%7D%3D%5B1%2C%5Cfrac%7B-1%2B%5Csqrt%7B5%7D+%7D%7B2%7D+%5D+%5E%7BT%7D%2C& alt=&\alpha _{1}=[1,\frac{1+\sqrt{5} }{2} ] ^{T},\alpha _{2}=[1,\frac{-1+\sqrt{5} }{2} ] ^{T},& eeimg=&1&&则&br&&br&&img src=&//www.zhihu.com/equation?tex=P%3D%5Cbegin%7Bbmatrix%7D%0A1%261+%5C%5C%0A%5Cfrac%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D+%26+%5Cfrac%7B%5Csqrt%7B5%7D-1%7D%7B2%7D%0A%5Cend%7Bbmatrix%7D%2C%0AP%5E%7B-1%7D%3D%5Cbegin%7Bbmatrix%7D%0A%5Cfrac%7B1-%5Csqrt%7B5%7D%7D%7B2%7D%261+%5C%5C%0A%5Cfrac%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D%26-1%0A%5Cend%7Bbmatrix%7D& alt=&P=\begin{bmatrix}
\frac{1+\sqrt{5}}{2} & \frac{\sqrt{5}-1}{2}
\end{bmatrix},
P^{-1}=\begin{bmatrix}
\frac{1-\sqrt{5}}{2}&1 \\
\frac{1+\sqrt{5}}{2}&-1
\end{bmatrix}& eeimg=&1&&&br&而&br&&img src=&//www.zhihu.com/equation?tex=%28PAP%5E%7B-1%7D%29%5E%7Bn%7D%3DPAP%5E%7B-1%7DPAP%5E%7B-1%7D...PAP%5E%7B-1%7D%3DPA%28P%5E%7B-1%7DP%29A%28P%5E%7B-1%7DP%29A...AP%5E%7B-1%7D%3DPA%5E%7Bn%7DP%5E%7B-1%7D& alt=&(PAP^{-1})^{n}=PAP^{-1}PAP^{-1}...PAP^{-1}=PA(P^{-1}P)A(P^{-1}P)A...AP^{-1}=PA^{n}P^{-1}& eeimg=&1&&&br&解出&img src=&//www.zhihu.com/equation?tex=A%5E%7Bn%7D%3DP%5E%7B-1%7D%28PAP%5E%7B-1%7D%29%5E%7Bn%7DP& alt=&A^{n}=P^{-1}(PAP^{-1})^{n}P& eeimg=&1&&,中间矩阵的n次方可以直接求出来：&br&&br&&img src=&//www.zhihu.com/equation?tex=%5Cbegin%7Bbmatrix%7D%0A%5Cfrac+%7B1-%5Csqrt%7B5%7D%7D%7B2%7D%260+%5C%5C%0A0%26%5Cfrac+%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D%0A%5Cend%7Bbmatrix%7D%5E%7Bn%7D%3D%0A%5Cbegin%7Bbmatrix%7D%0A%28%5Cfrac+%7B1-%5Csqrt%7B5%7D%7D%7B2%7D%29%5E%7Bn%7D%260%5C%5C%0A0%26%28%5Cfrac+%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D%29%5En%0A%5Cend%7Bbmatrix%7D& alt=&\begin{bmatrix}
\frac {1-\sqrt{5}}{2}&0 \\
0&\frac {1+\sqrt{5}}{2}
\end{bmatrix}^{n}=
\begin{bmatrix}
(\frac {1-\sqrt{5}}{2})^{n}&0\\
0&(\frac {1+\sqrt{5}}{2})^n
\end{bmatrix}& eeimg=&1&&&br&&br&然后可以轻易得到通项公式，上边已经给出，这里不再赘述。&br&&br&（3）特征方程解法&br&&br&通过方法（2），我们可以推导出一般的线性递推数列的通项求解方法，也就是特征方程法。我们可以发现，对于这种数列，通项总是可以表示为&img src=&//www.zhihu.com/equation?tex=f%28n%29%3DC_%7B1%7D+%5Clambda+_%7B1%7D%5E%7Bn%7D+%2BC_%7B2%7D+%5Clambda+_%7B2%7D%5E%7Bn%7D& alt=&f(n)=C_{1} \lambda _{1}^{n} +C_{2} \lambda _{2}^{n}& eeimg=&1&&的形式，因此可以直接利用已知项求解&img src=&//www.zhihu.com/equation?tex=C_%7B1%7D& alt=&C_{1}& eeimg=&1&&，&img src=&//www.zhihu.com/equation?tex=C_%7B2%7D+& alt=&C_{2} & eeimg=&1&&。具体做法如下：&br&&br&a.由递推数列构造特征方程&img src=&//www.zhihu.com/equation?tex=x%5E%7B2%7D%3Dx%2B1& alt=&x^{2}=x+1& eeimg=&1&&，解出两个特征值&img src=&//www.zhihu.com/equation?tex=%5Clambda_%7B1%7D%3D%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D+%2C%0A%5Clambda_%7B2%7D%3D%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D++& alt=&\lambda_{1}=\frac{1-\sqrt{5} }{2} ,
\lambda_{2}=\frac{1+\sqrt{5} }{2}
& eeimg=&1&&。&br&&br&b.带入&img src=&//www.zhihu.com/equation?tex=f%280%29%2Cf%281%29& alt=&f(0),f(1)& eeimg=&1&&,列出如下方程：&br&&br&&img src=&//www.zhihu.com/equation?tex=C_%7B1%7D+%2BC_%7B1%7D+%3D0& alt=&C_{1} +C_{1} =0& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D+C_%7B1%7D+%2B%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D+C_%7B2%7D+%3D1& alt=&\frac{1-\sqrt{5} }{2} C_{1} +\frac{1+\sqrt{5} }{2} C_{2} =1& eeimg=&1&&&br&&br&解得&img src=&//www.zhihu.com/equation?tex=C_%7B1%7D+%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+%2CC_%7B2%7D+%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+.& alt=&C_{1} =-\frac{1}{\sqrt{5} } ,C_{2} =\frac{1}{\sqrt{5} } .& eeimg=&1&&这样直接写出通项公式，是比较简单的做法。&br&&br&（4）母函数法（此方法涉及组合数学知识）&br&&br&设斐波那契数列的母函数为&img src=&//www.zhihu.com/equation?tex=G%28x%29& alt=&G(x)& eeimg=&1&&,即&br&&img src=&//www.zhihu.com/equation?tex=G%28x%29%3DF_%7B0%7D+%2BF_%7B1%7Dx%2BF_%7B2%7Dx%5E%7B2%7D%2BL%2BF_%7Bn%7Dx%5E%7Bn%7D& alt=&G(x)=F_{0} +F_{1}x+F_{2}x^{2}+L+F_{n}x^{n}& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%3D1%2Bx%2B%28F_%7B0%7D%2BF_%7B1%7D%29x%5E%7B2%7D%2BL%2B%28F_%7Bn-1%7D+%2BF_%7Bn-2%7D%29x%5E%7Bn%7D& alt=&=1+x+(F_{0}+F_{1})x^{2}+L+(F_{n-1} +F_{n-2})x^{n}& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%3D1%2B%28x%2BF_%7B1%7Dx%5E%7B2%7D%2BF_%7B2%7Dx%5E%7B3%7D%2BL%2B%29%2B%28F_%7B0%7Dx%5E%7B2%7D%2BF_%7B1%7Dx%5E3%2BL%29& alt=&=1+(x+F_{1}x^{2}+F_{2}x^{3}+L+)+(F_{0}x^{2}+F_{1}x^3+L)& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%3D1%2B%28x%2Bx%5E%7B2%7D%29G%28x%29& alt=&=1+(x+x^{2})G(x)& eeimg=&1&&&br&解得&img src=&//www.zhihu.com/equation?tex=G%28x%29%3D%5Cfrac%7B1%7D%7B1-x-x%5E%7B2%7D%7D+%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7D%5Cfrac%7B1%7D%7B1-%5Cfrac%7B1%2B%5Csqrt%7B5%7D+%7D%7B2%7Dx%7D+-%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D+%7D+%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7D%5Cfrac%7B1%7D%7B1-%5Cfrac%7B1-%5Csqrt%7B5%7D+%7D%7B2%7Dx%7D& alt=&G(x)=\frac{1}{1-x-x^{2}} =\frac{1}{\sqrt{5} } \frac{1+\sqrt{5} }{2}\frac{1}{1-\frac{1+\sqrt{5} }{2}x} -\frac{1}{\sqrt{5} } \frac{1-\sqrt{5} }{2}\frac{1}{1-\frac{1-\sqrt{5} }{2}x}& eeimg=&1&&&br&&br&再由幂级数展开公式&img src=&//www.zhihu.com/equation?tex=%5Cfrac%7B1%7D%7B1-x%7D%3D1%2Bx%2Bx%5E%7B2%7D%2B& alt=&\frac{1}{1-x}=1+x+x^{2}+& eeimg=&1&&……&br&&br&合并同类项并与&img src=&//www.zhihu.com/equation?tex=G%28x%29& alt=&G(x)& eeimg=&1&&的系数比较即可。&br&&br&&br&&br&到这里，求解斐波那契数列通项的方法就说的差不多了。无论是计算机求解还是数学推导，都体现出了非常多的技巧。而斐波那契数列的许多特性，就更加有意思了。&br&&br&&b&三、斐波那契数列的数学性质&/b&&br&&br&1.与黄金比的关系&br&&br&由通项公式，求相邻两项的商的极限，结果是黄金比，所以斐波那契数列又称为黄金比数列。斐波那契数列和黄金比还和一个有趣的数学概念——连分数有关：&br&&figure&&img src=&https://pic1.zhimg.com/50/bccfc83e3913_b.jpg& data-rawwidth=&355& data-rawheight=&139& class=&content_image& width=&355&&&/figure&2.一些简单的规律&br&&br&（1）任意连续四个斐波那契数，可以构造出一个毕达哥拉斯三元组。如取1,1,2,3.&br&中间两数相乘再乘2 ==》 4&br&外层2数乘积==》3&br&中间两数平方和==》5&br&得到3,4,5.&br&下一组是5,12,13,，有兴趣的读者可以再试着推一推，证明也是容易的。&br&&br&（2）整除性&br&&br&&p&每3个连续的斐波那契数有且只有一个被2整除，&/p&&br&&p&每4个连续的斐波那契数有且只有一个被3整除，&/p&&br&&p&每5个连续的斐波那契数有且只有一个被5整除，&/p&&br&&p&每6个连续的斐波那契数有且只有一个被8整除，&/p&&br&&p&每7个连续的斐波那契数有且只有一个被13整除，&/p&&p&…………&/p&&p&每n个连续的斐波那契数有且只有一个被&img src=&//www.zhihu.com/equation?tex=f%28n%29& alt=&f(n)& eeimg=&1&&整除.&br&&/p&&br&&p&（3）一些恒等式&/p&&br&&figure&&img src=&https://pic2.zhimg.com/50/f004b9eada9b8101ffa3_b.jpg& data-rawwidth=&532& data-rawheight=&519& class=&origin_image zh-lightbox-thumb& width=&532& data-original=&https://pic2.zhimg.com/50/f004b9eada9b8101ffa3_r.jpg&&&/figure&&br&&br&&p&3.杨辉三角中的斐波那契数列&/p&&br&&p&如图所示，每条斜线上的数的和就构成斐波那契数列。&/p&&p&&figure&&img src=&https://pic4.zhimg.com/50/dd81f7f0ab92f_b.jpg& data-rawwidth=&690& data-rawheight=&381& class=&origin_image zh-lightbox-thumb& width=&690& data-original=&https://pic4.zhimg.com/50/dd81f7f0ab92f_r.jpg&&&/figure&&br&即有&img src=&//www.zhihu.com/equation?tex=f%28n%29%3DC_%7Bn-1%7D%5E%7B0%7D+%2BC_%7Bn-2%7D%5E%7B1%7D%2BL%2BC_%7Bn-1-m%7D%5E%7Bm%7D& alt=&f(n)=C_{n-1}^{0} +C_{n-2}^{1}+L+C_{n-1-m}^{m}& eeimg=&1&&&/p&&br&&p&4.相关数列：卢卡斯（Lucas）数列&/p&&br&&p&卢卡斯数列的定义除了第0项为2之外，与斐波那契数列完全一致。即&/p&&figure&&img src=&https://pic1.zhimg.com/50/7bba6df1b75c06ceae49_b.jpg& data-rawwidth=&447& data-rawheight=&93& class=&origin_image zh-lightbox-thumb& width=&447& data-original=&https://pic1.zhimg.com/50/7bba6df1b75c06ceae49_r.jpg&&&/figure&&br&&p&其通项公式为：&/p&&figure&&img src=&https://pic1.zhimg.com/50/29b3b6e5e4cb75f83ba8f0_b.jpg& data-rawwidth=&323& data-rawheight=&85& class=&content_image& width=&323&&&/figure&&br&&p&卢卡斯数列和斐波那契数列有这些关系：&/p&&br&&img src=&//www.zhihu.com/equation?tex=F_%7B2n%7D%3DF_%7Bn%7DL_%7Bn%7D+& alt=&F_{2n}=F_{n}L_{n} & eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=5F_%7Bn%7D%3DL_%7Bn-1%7D%2BL_%7Bn%2B1%7D& alt=&5F_{n}=L_{n-1}+L_{n+1}& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=L_%7Bn%7D%5E%7B2%7D%3D5+F_%7Bn%7D%5E%7B2%7D%2B4%28-1%29%5E%7Bn%7D& alt=&L_{n}^{2}=5 F_{n}^{2}+4(-1)^{n}& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=%5Clim_%7Bn+%5Crightarrow+%5Cinfty+%7D%7B%5Cfrac%7BL_%7Bn%7D%7D%7BF_%7Bn%7D%7D+%7D+%3D%5Csqrt%7B5%7D+& alt=&\lim_{n \rightarrow \infty }{\frac{L_{n}}{F_{n}} } =\sqrt{5} & eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=L_%7Bn%7D%3DF_%7Bn-1%7D%2BF_%7Bn%2B1%7D& alt=&L_{n}=F_{n-1}+F_{n+1}& eeimg=&1&&&br&&br&&p&5.组合数学&/p&&br&&p&（1）一些恒等式&/p&&br&&figure&&img src=&https://pic3.zhimg.com/50/46aacf48efabbd3e9cb30_b.jpg& data-rawwidth=&482& data-rawheight=&481& class=&origin_image zh-lightbox-thumb& width=&482& data-original=&https://pic3.zhimg.com/50/46aacf48efabbd3e9cb30_r.jpg&&&/figure&&br&&p&（2）同余特性&/p&&br&&img src=&//www.zhihu.com/equation?tex=%28F%28m%29%2CF%28n%29%29%3D%28m%2Cn%29& alt=&(F(m),F(n))=(m,n)& eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=F%28m%29%7CF%28n%29%5CLeftrightarrow+m%7Cn& alt=&F(m)|F(n)\Leftrightarrow m|n& eeimg=&1&&&br&&p&当p为大于5的素数时，有：&/p&&img src=&//www.zhihu.com/equation?tex=F%28p-%28%5Cfrac%7Bp%7D%7B5%7D%29%29%5Cequiv+0%28modp%29+& alt=&F(p-(\frac{p}{5}))\equiv 0(modp) & eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=F%28p%29%5Cequiv+%28%5Cfrac%7Bp%7D%7B5%7D%29+%28modp%29+& alt=&F(p)\equiv (\frac{p}{5}) (modp) & eeimg=&1&&&br&&img src=&//www.zhihu.com/equation?tex=F%28p%2B%28%5Cfrac%7Bp%7D%7B5%7D%29%29%5Cequiv+1%28modp%29+& alt=&F(p+(\frac{p}{5}))\equiv 1(modp) & eeimg=&1&&&br&&p&其中&/p&&figure&&img src=&https://pic2.zhimg.com/50/c8f7b8c87987_b.jpg& data-rawwidth=&275& data-rawheight=&61& class=&content_image& width=&275&&&/figure&&br&&p&斐波那契数列还有许许多多的性质，我就不再一一介绍了。跑题了这么久，终于开始要真正回答问题了：斐波那契数列有什么用？&/p&&br&&p&&b&四、斐波那契数列的应用&/b&&/p&&br&&p&1.算法&/p&&p&a.斐波那契堆&/p&&br&&blockquote&&p&斐波那契堆(Fibonacci heap)是计算机科学中最小堆有序树的集合。它和二项式堆有类似的性质，可用于实现合并优先队列。特点是不涉及删除元素的操作有O(1)的平摊时间，用途包括稠密图每次Decrease－key只要O(1)的平摊时间，和二项堆的O(lgn)相比是巨大的改进。&br&&/p&&br&&p&斐波那契堆由一组最小堆构成，这些最小堆是有根的无序树。可以进行插入、查找、合并和删除等操作&/p&&p&1）插入：创建一个仅包含一个节点的新的斐波纳契堆，然后执行堆合并&/p&&p&2）查找：由于用一个指针指向了具有最小值的根节点，因此查找最小的节点是平凡的操作。&/p&&p&3）合并：简单合并两个斐波纳契堆的根表。即把两个斐波纳契堆的所有树的根首尾衔接并置。&/p&&p&4）删除（释放）最小节点&/p&&p&分为三步：&/p&&ol&&li&查找最小的根节点并删除它，其所有的子节点都加入堆的根表，即它的子树都成为堆所包含的树；&/li&&li&需要查找并维护堆的最小根节点，但这耗时较大。为此，同时完成堆的维护：对堆当前包含的树的度数从低到高，迭代执行具有相同度数的树的合并并实现最小树化调整，使得堆包含的树具有不同的度数。这一步使用一个数组，数组下标为根节点的度数，数组的值为指向该根节点指针。如果发现具有相同度数的其他根节点则合并两棵树并维护该数组的状态。&/li&&li&对当前堆的所有根节点查找最小的根节点。&/li&&/ol&5）降低一个点的键值：对一个节点的键值降低后，自键值降低的节点开始自下而上的迭代执行下述操作，直至到根节点或一个未被标记（marked）节点为止：&br&&p&如果当前节点键值小于其父节点的键值，则把该节点及其子树摘下来作为堆的新树的根节点；其原父节点如果是被标记（marked）节点，则也被摘下来作为堆的新树的根节点；如果其原父节点不是被标记（marked）节点且不是根节点，则其原父节点被加标记。&/p&&p&如果堆的新树的根节点被标记（marked），则去除该标记。&/p&&p&6）删除节点：把被删除节点的键值调整为负无穷小，然后执行“降低一个节点的键值”算法，然后再执行“删除最小节点”算法。&/p&&p&&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E6%E6%25B3%25A2%25E9%%25E5%25A5%%25A0%2586& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契堆&/a&&br&&/p&&/blockquote&&br&b.欧几里得算法的时间复杂度&br&&br&欧几里得算法是求解两个正整数最大公约数的算法，又称辗转相除法。代码如下：&br&&br&&div class=&highlight&&&pre&&code class=&language-c&&&span class=&kt&&int&/span& &span class=&nf&&gcd&/span&&span class=&p&&(&/span&&span class=&kt&&int&/span& &span class=&n&&a&/span&&span class=&p&&,&/span&&span class=&kt&&int&/span& &span class=&n&&b&/span&&span class=&p&&)&/span&
&span class=&p&&{&/span&
&span class=&k&&return&/span& &span class=&n&&b&/span& &span class=&o&&?&/span& &span class=&n&&gcd&/span&&span class=&p&&(&/span&&span class=&n&&b&/span&&span class=&p&&,&/span&&span class=&n&&a&/span&&span class=&o&&%&/span&&span class=&n&&b&/span&&span class=&p&&)&/span& &span class=&o&&:&/span& &span class=&n&&a&/span&&span class=&p&&;&/span&
&span class=&p&&}&/span&
&/code&&/pre&&/div&&br&在最坏的情况下，我们可以证明，若a较小，需要计算的次数为n，则&img src=&//www.zhihu.com/equation?tex=a%3EF%28n-1%29& alt=&a&F(n-1)& eeimg=&1&&.虽然说一般分析的时候会当成对数阶，但数论最常用的欧几里得算法竟然与斐波那契数列有关，也确实是很让人吃惊呢。&br&&br&2.物理学：氢原子能级问题&br&&br&&blockquote&&p&假定我们现在有一些氢气原子，一个电子最初所处的位置是最低的能级（Ground lever of energy），属于稳定状态。它能获得一个能量子或二个能量子（Quanta of energy）而使它上升到第一能级或者第二能级。但是在第一级的电子如失掉一个能量子就会下降到最低能级，它如获得一个能量子就会上升到第二级来。&/p&&br&&p&现在研究气体吸收和放出能量的情形，假定最初电子是处在稳定状态即零能级，然后让它吸收能量，这电子可以跳到第1能级或第2能级。然后再让这气体放射能量，这时电子在1级能级的就要下降到0能级，而在第2能级的可能下降到0能级或者第1能级的位置去。&/p&&br&&p&电子所处的状态可能的情形是：1、2、3、5、8、13、21…种。这是斐波那契数列的一部份。&br&&/p&&/blockquote&&br&3.自然界：植物的生长&br&&br&科学家发现，一些植物的花瓣、萼片、果实的数目以及排列的方式上，都有一个神奇的规律，它们都非常符合著名的斐波那契数列。例如：蓟，它们的头部几乎呈球状。在下图中，你可以看到两条不同方向的螺旋。我们可以数一下，顺时针旋转的（和左边那条旋转方向相同）螺旋一共有13条，而逆时针旋转的则有21条。此外还有菊花、向日葵、松果、菠萝等都是按这种方式生长的。&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/bb88e0bfc466_b.jpg& data-rawwidth=&298& data-rawheight=&293& class=&content_image& width=&298&&&/figure&&br&&figure&&img src=&https://pic4.zhimg.com/50/1c0bf51aa055bd675d5c28_b.jpg& data-rawwidth=&374& data-rawheight=&327& class=&content_image& width=&374&&&/figure&&br&&br&还有菠萝、松子等，也都符合这个特点，一般会出现34,55，89和144这几个数字。&br&&br&&figure&&img src=&https://pic4.zhimg.com/50/dc8bdf9d021a9fdea46f1b_b.jpg& data-rawwidth=&229& data-rawheight=&198& class=&content_image& width=&229&&&/figure&&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/cab9_b.jpg& data-rawwidth=&193& data-rawheight=&191& class=&content_image& width=&193&&&/figure&&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/30de68d1de6d4b86dc52eb_b.jpg& data-rawwidth=&199& data-rawheight=&190& class=&content_image& width=&199&&&/figure&&br&&br&最后上一张“斐波那契树”的图片：&br&&figure&&img src=&https://pic4.zhimg.com/50/6c4a54fba4c66143abb03c_b.jpg& data-rawwidth=&250& data-rawheight=&247& class=&content_image& width=&250&&&/figure&是的，这玩意就长这样，这种植物是存在的。&br&&br&4.波浪理论与股市&br&&br&这个答主不懂，大家可自行阅读文章&a href=&//link.zhihu.com/?target=http%3A//www.sbo8.com/bolanglilun/38440.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&波浪理论斐波那契数列与黄金分割率&/a&。&br&不过波浪的形状确实符合下边要说的斐波那契螺旋：&br&&figure&&img src=&https://pic3.zhimg.com/50/6b173ce40bc99eba839c5b_b.jpg& data-rawwidth=&484& data-rawheight=&443& class=&origin_image zh-lightbox-thumb& width=&484& data-original=&https://pic3.zhimg.com/50/6b173ce40bc99eba839c5b_r.jpg&&&/figure&&br&&br&5.斐波那契螺旋&br&&br&斐波那契螺旋又称黄金螺旋，在自然界中广泛存在。如图是一个边长为斐波那契数列的正方形组成的矩形。&br&&figure&&img src=&https://pic3.zhimg.com/50/0db4b080e5b_b.jpg& data-rawwidth=&223& data-rawheight=&138& class=&content_image& width=&223&&&/figure&&br&（加一句：看着这个图，是不是能发现&figure&&img src=&https://pic4.zhimg.com/50/bbc05fe2da6e397b9cecd7_b.jpg& data-rawwidth=&337& data-rawheight=&39& class=&content_image& width=&337&&&/figure&是显而易见的？）&br&&br&这样连起来就是斐波那契螺旋了&br&&figure&&img src=&https://pic3.zhimg.com/50/e7f6e0bfdfee5c52dcf791_b.jpg& data-rawwidth=&223& data-rawheight=&138& class=&content_image& width=&223&&&/figure&&br&&br&贝壳螺旋轮廓线&br&&figure&&img src=&https://pic2.zhimg.com/50/977bdf99ace9c519b3da7d7_b.jpg& data-rawwidth=&963& data-rawheight=&365& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic2.zhimg.com/50/977bdf99ace9c519b3da7d7_r.jpg&&&/figure&&br&&br&&figure&&img src=&https://pic4.zhimg.com/50/be079bf49c6_b.jpg& data-rawwidth=&963& data-rawheight=&577& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic4.zhimg.com/50/be079bf49c6_r.jpg&&&/figure&&figure&&img src=&https://pic2.zhimg.com/50/3903ae1febca8eae63f038aacd631564_b.jpg& data-rawwidth=&963& data-rawheight=&723& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic2.zhimg.com/50/3903ae1febca8eae63f038aacd631564_r.jpg&&&/figure&&br&&br&&br&&figure&&img src=&https://pic3.zhimg.com/50/7312cfbc09ead2a2b570f063bd0c2af5_b.jpg& data-rawwidth=&963& data-rawheight=&723& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic3.zhimg.com/50/7312cfbc09ead2a2b570f063bd0c2af5_r.jpg&&&/figure&&br&&figure&&img src=&https://pic1.zhimg.com/50/0f99eeda9bf_b.jpg& data-rawwidth=&963& data-rawheight=&555& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic1.zhimg.com/50/0f99eeda9bf_r.jpg&&&/figure&&br&向日葵的生长&br&&figure&&img src=&https://pic3.zhimg.com/50/a40a8ec6d3bb9_b.jpg& data-rawwidth=&963& data-rawheight=&723& class=&origin_image zh-lightbox-thumb& width=&963& data-original=&https://pic3.zhimg.com/50/a40a8ec6d3bb9_r.jpg&&&/figure&&br&神奇的花&br&&figure&&img src=&https://pic2.zhimg.com/50/db5d355d805ba7f96b576ac2a5ed1982_b.jpg& data-rawwidth=&723& data-rawheight=&723& class=&origin_image zh-lightbox-thumb& width=&723& data-original=&https://pic2.zhimg.com/50/db5d355d805ba7f96b576ac2a5ed1982_r.jpg&&&/figure&&br&&br&6.建筑学&br&&br&&figure&&img src=&https://pic4.zhimg.com/50/27d8c5e1cdf8e107a3752a_b.jpg& data-rawwidth=&600& data-rawheight=&400& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic4.zhimg.com/50/27d8c5e1cdf8e107a3752a_r.jpg&&&/figure&&br&&figure&&img src=&https://pic4.zhimg.com/50/8e5b641cadfdfe793c43a71edbbfd582_b.jpg& data-rawwidth=&490& data-rawheight=&367& class=&origin_image zh-lightbox-thumb& width=&490& data-original=&https://pic4.zhimg.com/50/8e5b641cadfdfe793c43a71edbbfd582_r.jpg&&&/figure&&br&&figure&&img src=&https://pic2.zhimg.com/50/e400a2c0dbfd90c_b.jpg& data-rawwidth=&580& data-rawheight=&359& class=&origin_image zh-lightbox-thumb& width=&580& data-original=&https://pic2.zhimg.com/50/e400a2c0dbfd90c_r.jpg&&&/figure&&br&&figure&&img src=&https://pic2.zhimg.com/50/6dc0cda9399d53ccaee6c1_b.jpg& data-rawwidth=&370& data-rawheight=&500& class=&content_image& width=&370&&&/figure&&br&7.据说一个小男孩参考斐波那契数列发明了太阳能电池树：&br&&br&&blockquote&一名13岁的男孩根据斐波那契数列&a href=&//link.zhihu.com/?target=http%3A//inhabitat.com/13-year-old-makes-solar-power-breakthrough-by-harnessing-the-fibonacci-sequence/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&发明了太阳能电池树&/a&，其产生的电力比太阳能光伏电池阵列多20-50%。&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E6%E6%25B3%25A2%25E9%%25E5%25A5%%%25E5%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列&/a&类似从0和1开始，之后的数是之前两数的和，如0，1，1，2，3，5，8，13，21...&a href=&//link.zhihu.com/?target=http%3A//www.amnh.org/nationalcenter/youngnaturalistawards/2011/aidan.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Aidan Dwye&/a&在观察树枝分叉时发现它的分布模式类似斐波那契数列，这是大自然演化的一种结果，可能有助于树叶进行光合作用。&br&因此，Dwye猜想为什么不按照斐波那契数列排列太阳能电池？他设计了太阳能电池树，发现它的输出电力提高了20%，每天接受光照的时间延长了2.5小时。&/blockquote&&br&8.斐波那契螺旋形的摇椅&br&&br&&blockquote&妈妈摇椅是设计师Patrick Messier为自己的妻子兼合作伙伴Sophie Fournier设计的，当时他们刚有了第一个宝宝。&br&&p&当Sophie宣布自己怀孕时，她说想要一把摇椅，但发现没有一把摇椅能满足美观舒适的标准，于是Patrick决定自己做一把。&/p&&p&于是就有了这把妈妈摇椅。像是一个飘在空中的丝带，由一片纤维玻璃做成，曲线服从&a href=&//link.zhihu.com/?target=http%3A//www.shejipi.com/2748.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列分布&/a&，经过特殊的高光聚氨酯处理。&/p&&/blockquote&&figure&&img src=&https://pic2.zhimg.com/50/7b83bf1c98af7dec0cf992_b.jpg& data-rawwidth=&600& data-rawheight=&388& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic2.zhimg.com/50/7b83bf1c98af7dec0cf992_r.jpg&&&/figure&&br&&br&&b&五、数学上的扩展&/b&&br&&br&（1）广义斐波那契数列&br&定义：&img src=&//www.zhihu.com/equation?tex=a%2Bb%2Cab%5Cin+Z& alt=&a+b,ab\in Z& eeimg=&1&&,数列&img src=&//www.zhihu.com/equation?tex=f%28n%29& alt=&f(n)& eeimg=&1&&满足：&br&&figure&&img src=&https://pic2.zhimg.com/50/aee0c8f3fb2be52c05fa1f8d4b2857d2_b.jpg& data-rawwidth=&326& data-rawheight=&95& class=&content_image& width=&326&&&/figure&其通项为：&br&&figure&&img src=&https://pic2.zhimg.com/50/452e6b08c4a063a6430b8_b.jpg& data-rawwidth=&117& data-rawheight=&57& class=&content_image& width=&117&&&/figure&当&img src=&//www.zhihu.com/equation?tex=a%2Bb%3D1%2Cab%3D-1& alt=&a+b=1,ab=-1& eeimg=&1&&时即为斐波那契数列。&br&&br&（2）反斐波那契数列&br&&br&定义：&img src=&//www.zhihu.com/equation?tex=g%28n%2B2%29%3Dg%28n%29-g%28n%2B1%29& alt=&g(n+2)=g(n)-g(n+1)& eeimg=&1&&&br&反斐波那契数列相邻项比值的极限为&img src=&//www.zhihu.com/equation?tex=-0.618& alt=&-0.618& eeimg=&1&&。&br&&br&（3）巴都万数列(&a href=&//link.zhihu.com/?target=http%3A//oeis.org/A000931& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&A000931 - OEIS&/a&)&br&斐波那契数列可以刻画矩形，而巴都万数列则刻画的是三角形。其定义如下：&br&&figure&&img src=&https://pic2.zhimg.com/50/dcdfbcaf5f4b_b.jpg& data-rawwidth=&211& data-rawheight=&68& class=&content_image& width=&211&&&/figure&&br&(4)未解之谜：角谷猜想&br&&br&对一个正整数，若为奇数则乘3加1，若为偶数则除以2，通过有限次这样的操作，能否使得该数变成1？&br&这个猜想和斐波那契数列又很大关系，具体的可以看&a href=&//link.zhihu.com/?target=http%3A//www.doc88.com/p-5.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&角谷猜想的斐波那契数列现象&/a&。&br&&br&&br&&b&六、总结&/b&&br&&br&斐波那契数列是各个学科中都出现的小滑头，它许多漂亮的性质让我们着迷。上文我所描述的这些只是它的冰山一角，权当抛砖引玉。大家读完了我的答案，还可以再结合自己的专业去看一些相关的资料，更好的去了解这个有趣的数列。&br&&br&&b&七、参考文献&/b&&br&&br&&br&&br&&br&[1]&a href=&//link.zhihu.com/?target=http%3A//www.hytc.cn/xsjl/szh/lec5.pdf& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://www.&/span&&span class=&visible&&hytc.cn/xsjl/szh/lec5.p&/span&&span class=&invisible&&df&/span&&span class=&ellipsis&&&/span&&/a&&br&[2]&a href=&//link.zhihu.com/?target=http%3A//blog.sina.com.cn/s/blog_77eb54a40100rpp9.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列_一米阳光&/a&&br&[3]&a href=&//link.zhihu.com/?target=http%3A//www.360doc.com/content/14/02.shtml& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列的规律性&/a&&br&[4]&a href=&//link.zhihu.com/?target=http%3A//www.cnbeta.com/articles/152349.htm& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&13岁男孩根据斐波那契数列发明太阳能电池树_cnBeta 人物_cnBeta.COM&/a&&br&[5]&a href=&//link.zhihu.com/?target=http%3A//www.shejipi.com/4547.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&服从斐波那契数列分布的妈妈摇椅&/a&&br&[6]&a href=&//link.zhihu.com/?target=http%3A//shiba.hpe.sh.cn/jiaoyanzu/wuli/ShowArticle.aspx%3FarticleId%3D990%26classId%3D5& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&从斐波那契数列到黄金分割&/a&&br&[7]&a href=&//link.zhihu.com/?target=http%3A//max.book118.com/html/82822.shtm& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契《计算之书》的研究.pdf 全文 文档投稿网&/a&&br&[8]&a href=&//link.zhihu.com/?target=http%3A//www.cnblogs.com/newpanderking/archive//2117328.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&斐波那契数列&/a&&br&['9]&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/zh-tw/%25E6%E6%25B3%25A2%25E9%%25E5%25A5%%%25E5%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&費氏數列&/a&&br&[10]&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E5%25B7%25B4%25E9%2583%25BD%25E8%2590%25AC%25E6%%25E5%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&巴都萬數列&/a&&br&[11]&a href=&//link.zhihu.com/?target=http%3A//zh.wikipedia.org/wiki/%25E5%258D%25A2%25E5%258D%25A1%25E6%2596%25AF%25E6%& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&zh.wikipedia.org/wiki/%&/span&&span class=&invisible&&E5%8D%A2%E5%8D%A1%E6%96%AF%E6%95%B0&/span&&span class=&ellipsis&&&/span&&/a&&br&[12]&a href=&//link.zhihu.com/?target=http%3A//www.doc88.com/p-5.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&角谷猜想的斐波那契数列现象&/a&&br&[13]&a href=&//link.zhihu.com/?target=http%3A//oeis.org/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&The On-Line Encyclopedia of Integer Sequences(R) (OEIS(R))&/a&&br&[14]&a href=&//link.zhihu.com/?target=http%3A//www.sbo8.com/bolanglilun/38440.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&波浪理论斐波那契数列与黄金分割率&/a&
一句话先回答问题：因为斐波那契数列在数学和生活以及自然界中都非常有用。 下面我就尽我所能，讲述一下斐波那契数列。 一、起源和定义 斐波那契数列最早被提出是印度数学家Gopala，他在研究箱子包装物件长度恰好为1和2时的方法数时首先描述了这个数列。也…
刚才翻官网时找到官方答案了，顺便把其他人的也都放上来：&br&&a href=&//link.zhihu.com/?target=http%3A//piapro.net/intl/zh-cn_character.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&piapro.net&/a&&br&&br&MIKU &br&苍绿色 #39C5BB rgb(57, 197, 187) &figure&&img src=&https://pic1.zhimg.com/50/v2-b0dfda47f94fcb5234b5f_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&RIN &br&橙黄色 #FFA500 rgb(255, 165, 0)&br&&figure&&img src=&https://pic2.zhimg.com/50/v2-d1d85b885ea0fdb34f4bfde9_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&LEN &br&黄色 #FFE212 rgb(255, 226, 17)&figure&&img src=&https://pic4.zhimg.com/50/v2-9462eeebcace77d84ee60_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&LUKA &br&粉色 #FFBFCB rgb(255, 192, 203)&figure&&img src=&https://pic1.zhimg.com/50/v2-6c3acd9c65fc7c3c8bb0d_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&MEIKO &br&红色 #D80000 rgb(216, 0, 0)&figure&&img src=&https://pic4.zhimg.com/50/v2-844fbf323219ecfdd98bbec_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&&br&KAITO &br&蓝色 #0000FF rgb(0, 0, 255)&figure&&img src=&https://pic3.zhimg.com/50/v2-6a1e411f3e1ff58e3d4388_b.jpg& data-rawwidth=&20& data-rawheight=&20& class=&content_image& width=&20&&&/figure&
刚才翻官网时找到官方答案了，顺便把其他人的也都放上来：
MIKU 苍绿色 #39C5BB rgb(57, 197, 187) RIN 橙黄色 #FFA500 rgb(255, 165, 0) LEN 黄色 #FFE212 rgb(255, 226, 17) LUKA 粉色 #FFBFCB rgb(255, 192, 203) MEIKO 红色 #D80000 rgb(216…
金句王。肥皂终结者。&br&&br&&b&壹【金句王】 &/b&&br& (转载自豆瓣【整理+重新翻译和润色 by 慕容澈，注：破折号之后的是不同的译本。】 ）&br&&br&The only difference between a caprice and a life-long passion is that the caprice lasts a little longer. &br&逢场作戏和终身不渝之间的区别只在于逢场作戏稍微长一些。 &br&&br&When one is in love, one always begins by deceiveing one's self, and one always ends by deceiving others. That is what would calls a romance. &br&爱，始于自我欺骗，终于欺骗他人。这就是所谓的浪漫。 &br&——恋爱总是以自欺开始，以欺人结束。 &br&&br&The very essence of romance is uncertainty. &br&浪漫的精髓就在于它充满种种可能。 &br&&br&Man is a rational animal who always loses his temper when he is called upon to act in accordance with the dictates of reason. &br&人是理性动物，但当他被要求按照理性的要求行动时，可又要发脾气了。 &br&&br&My wallpaper and I are fighting a duel to the death. One or other of us has to go. &br&墙纸越来越破，而我越来越老，两者之间总有一个要先消失。——日，于左岸旅店，他的遗言。 &br&&br&No man is rich enough to buy back his own past. &br&-An Ideal Husband (1895) &br&没有人富有到可以赎回自己的过去。 &br&——改编自《理想的丈夫》 &br&&br&The truth is rarely pure and never simple. &br&-The Importance of Being Earnest (1895) &br&真相很少纯粹，也决不简单。 &br&——《不可儿嬉》 &br&&br&To love oneself is the beginning of a lifelong romance. &br&爱自己是终身浪漫的开始。 &br&——爱自己就是开始延续一生的罗曼史。 &br&&br&We are all in the gutter, but some of us are looking at the stars. &br&我们都生活在阴沟里，但仍有人仰望星空。 &br&&br&Most people discover when it is too late that the only things one never regrets are one's mistakes. &br&大多数人发现他们从未后悔的事情只是他们的错误，但发现时已经太晚了。 &br&&br&What is the chief cause of divorce? Marriage. &br&什么是离婚的主要原因？结婚。 &br&&br&When a love comes to an end, weaklings cry, efficient ones instantly find another love and wise already had one in reserve. &br&当爱情走到尽头，软弱者哭个不停，有效率的马上去寻找下一个目标，而聪明的早就预备了下一个。 &br&&br&No great artist ever sees things as they really are. If he did he would cease to be an artist.&br&伟大的艺术家所看到的，从来都不是世界的本来面目。一旦他看透了，他就不再是艺术家。&br&&br&I represent to you all the sins you have never had the courage to commit. &br&我给你们讲述的是所有你们没勇气去犯的罪孽。 &br&&br&One can always be kind to people one cares nothing about. &br&一个人总是可以善待他毫不在意的人。 &br&&br&We Irish are too
we are a nation of brilliant failures, but we are the greatest talkers since the Greeks. &br&我们爱尔兰人太诗意以至不能做诗人，我们的国家里充满才华横溢的失败者，可我们是自希腊人以来最伟大的说空话之人。 &br&&br&What seems to us as bitter trials are often blessings in disguise &br&看似痛苦的试炼的往往是伪装的祝福。 &br&&br&The advantage of the emotions is that they lead us astray. &br&情感的好处就是让我们误入歧途。 &br&&br&Over the piano was printed a notice: Please do not shoot the pianist. He is doing his best.&br&Personal Impressions of America (Leadville) (1883) &br&钢琴上贴着一条告示：请不要枪杀钢琴师，他已经尽力了。 &br&&br&The heart was made to be broken. &br&心是用来碎的。 &br&&br&The public is wonderfully tolerant. It forgives everything except genius. &br&公众惊人地宽容。他们可以原谅一切，除了天才。 &br&&br&Religions die when they are proved to be true. Science is the record of dead religions. &br&宗教一旦被证明是正确时就会消亡。科学便是已消亡宗教的记录。 &br&&br&Why was I born with such contemporaries &br&为什么我会和这样同时代的人一块出生呢？ &br&&br&A poet can survive everything but a misprint. &br&诗人可以从任何事件中存活，印刷错误除外。 &br&&br&Only the shallow know themselves. &br&-Phrases and Philosophies for the use of the Young (1894) &br&只有浅薄的人才了解自己。 &br&&br&The only way to get rid of temptation is to yield to it... I can resist everything but temptation. &br&摆脱诱惑的唯一方式是臣服于诱惑……我能抗拒一切，除了诱惑。 &br&&br&Discontent is the first step in the progress of a man or a nation &br&不满是个人或民族迈向进步的第一步。 &br&&br&I like to do all the talking myself. It saves time, and prevents arguments. &br&我喜欢自言自语，因为这样节约时间，而且不会有人跟我争论。 &br&&br&Quotation is a serviceable substitute for wit. &br&格言是智慧耐用的替代品。 &br&&br&A dreamer is one who can only find his way by moonlight, and his punishment is that he sees the dawn before the rest of the world. &br&梦想家只能在月光下找到前进的方向，他为此遭受的惩罚是比所有人提前看到曙光。 &br&&br&Every saint has a past and every sinner has a future. &br&每个圣人都有过去，每个罪人都有未来。 &br&&br&To live is the rarest thing in the world. Most people exist, that is all. &br&生活是世上最罕见的事情，大多数人只是存在，仅此而已。 &br&&br&I have nothing to declare except my genius. &br&除了我的天才，我没什么好申报的。 &br&&br&I like men who have a future and women who have a past &br&我喜欢有未来的男人和有过去的女人。 &br&&br&Pessimist: One who, when he has the choice of two evils, chooses both. &br&悲观主义者是这种人：当他可以从两种罪恶中选择时，他把两种都选了。 &br&&br&Society exists only in the real world there are only individuals. &br&社会仅仅以一种精神概念而存在，真实世界中只有个体存在。 &br&&br&What is a cynic? A man who knows the price of everything and the value of nothing &br&一个愤世嫉俗的人知道所有东西的价格，却不知道任何东西的价值。 &br&&br&I like persons better than principles, and I like persons with no principles better than anything else in the world. &br&我喜欢人甚于原则，此外我还喜欢没原则的人甚于世界上的一切。 &br&&br&It is the confession, not the priest, that gives us absolution &br&给我们赦免的，是忏悔而不是牧师。 &br&&br&I don't wa I want to live. &br&我不想谋生；我想生活。 &br&&br&Moderation is a fatal thing. Nothing succeeds like excess. &br&适度是极其致命的事情。过度带来的成功是无可比拟的。 &br&&br&The only thing worse than being talked about is not being talked about. &br&世上只有一件事比被人议论更糟糕，那就是没有人议论你。 &br&&br&When the gods wish to punish us, they answer our prayers &br&-An Ideal husband, 1893 &br&当神想惩罚我们时，他们就回应我们的祈祷。 &br&-《一个理想的丈夫》 &br&&br&&Life is never fair...And perhaps it is a good thing for most of us that it is not.& &br&-An Ideal Husband，1893 &br&生活从来不是公平的……而且，或许对我们大多数人来说，这是件好事。 &br&-《一个理想的丈夫》 &br&&br&How can a woman be expected to be happy with a man who insists on treating her as if she were a perfectly normal human being. &br&如果男人坚持把她当作一个完全正常的人，女人如何能期望会从他那里获得幸福。 &br&&br&All charming people, I fancy, are spoiled. It is the secret of their attraction &br&我想所有迷人的人都是被宠爱着的，这是他们吸引力来源的秘密。 &br&&br&Nothing is so aggravating than calmness. &br&没有比冷静更让人恼火的。 &br&&br&Popularity is the one insult I have never suffered. &br&声望是我从未经受的侮辱之一。 &br&&br&Ridicule is the tribute paid to the genius by the mediocrities. &br&奚落是庸才对天才的颂歌。 &br&&br&To do nothing at all is the most difficult thing in the world, the most difficult and the most intellectual &br&什么也不做是世上最难的事情，最困难并且最智慧。 &br&&br&A true friend stabs you in the front. &br&真朋友才会当面中伤你。 &br&&br&Ordinary riches can be stolen, real riches cannot. In your soul are infinitely precious things that cannot be taken from you. &br&平常的财宝会被偷走，而真正的财富则不会。你灵魂里无限珍贵的东西是无法被夺走的。 &br&&br&The well bred contradict other people. The wise contradict themselves. &br&教养良好的人处处和他人过不去，头脑聪明的人处处和自己过不去。 &br&&br&Hatred is blind, as well as love. &br&恨是盲目的，爱亦然。 &br&&br&Looking good and dressing well is a necessity. Having a purpose in life is not. &br&注意穿着打扮是必要的。而拥有生活目标却并非如此。 &br&&br&Always forgive your enemies - nothing annoys them so much. &br&永远宽恕你的敌人，没有什么能比这个更让他们恼怒的了。 &br&&br&Children begin by
as they grow o sometimes, they forgive them &br&孩子最初爱他们父母，等大一些他们评判父母；然后有些时候，他们原谅父母。 &br&&br&There are only two tragedies in life: one is not getting what one wants, and the other is getting it. &br&生活中只有两种悲剧:一个是没有得到我们想要的，另外一个是得到我们想要的。 &br&——世上只有两类悲剧，有些人总是不能遂愿，而有些人总是心想事成。 &br&&br&A gentleman is one who never hurts anyone's feelings unintentionally. &br&绅士就是从不无心伤害别人感觉的人。 &br&&br&Conversation about the weather is the last refuge of the unimaginative &br&谈论天气是无趣人类最后的避难所。 &br&&br&One's real life is often the life that one does not lead. &br&真实生活就通常就是我们无法掌控的生活。 &br&&br&Wickedness is a myth invented by good people to account for the curious attractiveness of others &br&邪恶是善良的人们编造的谎言，用来解释其他人的特殊魅力。 &br&&}
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