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F⁡(n)&#x3D;55⋅[(1+5)n−(1−5)n]\operatorname{F}(n)&#x3D;\dfrac{\sqrt{5}}{5}\cdot\left[\left(1+\sqrt{5}\right)^n-\left(1-\sqrt{5}\right)^n\right] F(n)&#x3D;55⋅[(1+5)n−(1−5)n] 代码揍这么简单： 123int"><meta property="og:type" content="article"><meta property="og:title" content="斐波那契数列-O(1)"><meta property="og:url" content="https://schtonn.github.io/posts/fibonacci/index.html"><meta property="og:site_name" content="Alex&#39;s Blog"><meta property="og:description" content="引自：《信息学奥赛之-数学一本通》 揍是这样： F⁡(n)&#x3D;55⋅[(1+5)n−(1−5)n]\operatorname{F}(n)&#x3D;\dfrac{\sqrt{5}}{5}\cdot\left[\left(1+\sqrt{5}\right)^n-\left(1-\sqrt{5}\right)^n\right] F(n)&#x3D;55⋅[(1+5)n−(1−5)n] 代码揍这么简单： 123int"><meta property="og:locale" content="en_US"><meta property="article:published_time" content="2020-03-02T03:35:44.000Z"><meta property="article:modified_time" content="2020-03-29T14:23:24.562Z"><meta property="article:author" content="Alex"><meta property="article:tag" content="math"><meta name="twitter:card" 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class="post-meta"><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-o"></i></span> <span class="post-meta-item-text">Posted on</span> <time title="Created: 2020-Mar-02 11:35:44" itemprop="dateCreated datePublished" datetime="2020-03-02T11:35:44+08:00">2020-Mar-02</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-check-o"></i></span> <span class="post-meta-item-text">Edited on</span> <time title="Modified: 2020-Mar-29 22:23:24" itemprop="dateModified" datetime="2020-03-29T22:23:24+08:00">2020-Mar-29</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-comment-o"></i></span> <span class="post-meta-item-text">Valine:</span><a title="valine" href="/posts/fibonacci/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/posts/fibonacci/" itemprop="commentCount"></span></a></span></div></header><div class="post-body" itemprop="articleBody"><p>引自：《信息学奥赛之-数学一本通》<br> 揍是这样：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">F</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>5</mn></mfrac><mo>⋅</mo><mrow><mo fence="true">[</mo><msup><mrow><mo fence="true">(</mo><mn>1</mn><mo>+</mo><msqrt><mn>5</mn></msqrt><mo fence="true">)</mo></mrow><mi>n</mi></msup><mo>−</mo><msup><mrow><mo fence="true">(</mo><mn>1</mn><mo>−</mo><msqrt><mn>5</mn></msqrt><mo fence="true">)</mo></mrow><mi>n</mi></msup><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\operatorname{F}(n)=\dfrac{\sqrt{5}}{5}\cdot\left[\left(1+\sqrt{5}\right)^n-\left(1-\sqrt{5}\right)^n\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mop"><span class="mord mathrm">F</span></span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.27022em;vertical-align:-.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5842200000000002em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.90722em"><span class="svg-align" style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord" style="padding-left:.833em"><span class="mord">5</span></span></span><span style="top:-2.86722em"><span class="pstrut" style="height:3em"></span><span class="hide-tail" style="min-width:.853em;height:1.08em"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95 702c-2.7 0-7.17-2.7-13.5-8-5.8-5.3-9.5-10-9.5-14 0-2 .3-3.3 1-4 1.3-2.7 23.83-20.7 67.5-54 44.2-33.3 65.8-50.3 66.5-51 1.3-1.3 3-2 5-2 4.7 0 8.7 3.3 12 10s173 378 173 378c.7 0 35.3-71 104-213 68.7-142 137.5-285 206.5-429 69-144 104.5-217.7 106.5-221 5.3-9.3 12-14 20-14H400000v40H845.2724s-225.272 467-225.272 467-235 486-235 486c-2.7 4.7-9 7-19 7-6 0-10-1-12-3s-194-422-194-422-65 47-65 47zM834 80H400000v40H845z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.13278em"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.854312em;vertical-align:-.65002em"></span><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size2">[</span></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size2">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.956095em"><span class="svg-align" style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord" style="padding-left:.833em"><span class="mord">5</span></span></span><span style="top:-2.916095em"><span class="pstrut" style="height:3em"></span><span class="hide-tail" style="min-width:.853em;height:1.08em"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95 702c-2.7 0-7.17-2.7-13.5-8-5.8-5.3-9.5-10-9.5-14 0-2 .3-3.3 1-4 1.3-2.7 23.83-20.7 67.5-54 44.2-33.3 65.8-50.3 66.5-51 1.3-1.3 3-2 5-2 4.7 0 8.7 3.3 12 10s173 378 173 378c.7 0 35.3-71 104-213 68.7-142 137.5-285 206.5-429 69-144 104.5-217.7 106.5-221 5.3-9.3 12-14 20-14H400000v40H845.2724s-225.272 467-225.272 467-235 486-235 486c-2.7 4.7-9 7-19 7-6 0-10-1-12-3s-194-422-194-422-65 47-65 47zM834 80H400000v40H845z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.08390500000000001em"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.204292em"><span style="top:-3.6029em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size2">(</span></span><span class="mord">1</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.956095em"><span class="svg-align" style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord" style="padding-left:.833em"><span class="mord">5</span></span></span><span style="top:-2.916095em"><span class="pstrut" style="height:3em"></span><span class="hide-tail" style="min-width:.853em;height:1.08em"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95 702c-2.7 0-7.17-2.7-13.5-8-5.8-5.3-9.5-10-9.5-14 0-2 .3-3.3 1-4 1.3-2.7 23.83-20.7 67.5-54 44.2-33.3 65.8-50.3 66.5-51 1.3-1.3 3-2 5-2 4.7 0 8.7 3.3 12 10s173 378 173 378c.7 0 35.3-71 104-213 68.7-142 137.5-285 206.5-429 69-144 104.5-217.7 106.5-221 5.3-9.3 12-14 20-14H400000v40H845.2724s-225.272 467-225.272 467-235 486-235 486c-2.7 4.7-9 7-19 7-6 0-10-1-12-3s-194-422-194-422-65 47-65 47zM834 80H400000v40H845z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.08390500000000001em"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.204292em"><span style="top:-3.6029em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size2">]</span></span></span></span></span></span></span></p><p>代码揍这么简单：</p><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">fibo</span><span class="params">(<span class="keyword">int</span> n)</span></span>&#123;</span><br><span class="line">	<span class="keyword">return</span> (<span class="built_in">sqrt</span>(<span class="number">5</span>)/<span class="number">5</span>)*(<span class="built_in">pow</span>((<span class="number">1</span>+<span class="built_in">sqrt</span>(<span class="number">5</span>))/<span class="number">2</span>,n)-<span class="built_in">pow</span>((<span class="number">1</span>-<span class="built_in">sqrt</span>(<span class="number">5</span>))/<span class="number">2</span>,n));</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure><p>要是再加个快速幂就更棒了（够快了，懒得写）</p></div><div><ul class="post-copyright"><li class="post-copyright-author"> <strong>Post author:</strong> Alex</li><li class="post-copyright-link"> <strong>Post link:</strong> <a href="https://schtonn.github.io/posts/fibonacci/" title="斐波那契数列-O(1)">https://schtonn.github.io/posts/fibonacci/</a></li><li class="post-copyright-license"> <strong>Copyright Notice:</strong> All articles in this blog are licensed under<a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="noopener" target="_blank"><i class="fa fa-fw fa-creative-commons"></i> BY-NC-SA</a> unless stating additionally.</li></ul></div><div class="followme"><p>Welcome to my other publishing channels</p><div class="social-list"><div class="social-item"><a target="_blank" class="social-link" href="/images/wechat_channel.jpg"><span class="icon"><i class="fa fa-wechat"></i></span> <span class="label">WeChat</span></a></div><div 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      visitor    : false,
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      path       : location.pathname,
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