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<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
<title>概述</title>
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<body class="vscode-body vscode-light">
<center>
<H1>No Sandwich Swap</H1>
</center>
<center>by WHU Web3 Club</center>
<h2 id="概述">概述</h2>
<p>这是一个抗 MEV 的 DEX ，使用<strong>双曲集合竞价</strong>的方式，将一段时间内的若干笔交易加总后无限细分并均匀交错排列，从而摊平了单笔交易的价格冲击效应，将 MEV
攻击者的交易分散至整个周期中使得套利收益大幅降低。</p>
<p>该 DEX 的意义在于：</p>
<ol>
<li>减少价格冲击，让实际成交价更加贴近公允价格</li>
<li>有效防范 MEV 攻击，使得三明治攻击变得无利可图</li>
<li>有效降低普通交易者遭遇的滑点，提高用户体验</li>
<li>抑制价格的波动性，减少 LP 遭遇的无常损失</li>
<li>通过治理代币激励机制，鼓励用户多多交易，提高市场流动性</li>
</ol>
<h2 id="原理">原理</h2>
<p>设期初流动性池满足 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
<mo>×</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>=</mo>
<mi>k</mi>
</mrow>
<annotation encoding="application/x-tex">x_0 \times y_0=k</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span
class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3011em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace"
style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace"
style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut"
style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3011em;"><span
style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace"
style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace"
style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut"
style="height:0.6944em;"></span><span class="mord mathnormal"
style="margin-right:0.03148em;">k</span></span></span></span> 。对于每一个结算周期，流动性池收到的基准货币和计价货币数量分别为 <span
class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mo stretchy="false">{</mo>
<msub>
<mi>a</mi>
<mn>1</mn>
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<mo separator="true">,</mo>
<msub>
<mi>a</mi>
<mn>2</mn>
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<mo separator="true">,</mo>
<mi mathvariant="normal">.</mi>
<mi mathvariant="normal">.</mi>
<mi mathvariant="normal">.</mi>
<mo separator="true">,</mo>
<msub>
<mi>a</mi>
<mi>A</mi>
</msub>
<mo stretchy="false">}</mo>
<mo separator="true">,</mo>
<mo stretchy="false">{</mo>
<msub>
<mi>b</mi>
<mn>1</mn>
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<mo separator="true">,</mo>
<msub>
<mi>b</mi>
<mn>2</mn>
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<mo separator="true">,</mo>
<mi mathvariant="normal">.</mi>
<mi mathvariant="normal">.</mi>
<mi mathvariant="normal">.</mi>
<mo separator="true">,</mo>
<msub>
<mi>b</mi>
<mi>B</mi>
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<mo stretchy="false">}</mo>
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<annotation encoding="application/x-tex">\{a_1, a_2, ..., a_A\}, \{b_1, b_2, ..., b_B\}</annotation>
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</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span
class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3011em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span
class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span
class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3011em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span
class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span
class="mord">...</span><span class="mpunct">,</span><span class="mspace"
style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"
style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span
class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">A</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span
class="mclose">}</span><span class="mpunct">,</span><span class="mspace"
style="margin-right:0.1667em;"></span><span class="mopen">{</span><span class="mord"><span
class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3011em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span
class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span
class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3011em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span
class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span
class="mord">...</span><span class="mpunct">,</span><span class="mspace"
style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span
class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"
style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span
class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight"
style="margin-right:0.05017em;">B</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span
class="mclose">}</span></span></span></span> ，设 <span class="katex"><span class="katex-mathml"><math
xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle scriptlevel="0" displaystyle="true">
<mi>α</mi>
<mo>=</mo>
<munderover>
<mo>∑</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>A</mi>
</munderover>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
<mo separator="true">,</mo>
<mi>β</mi>
<mo>=</mo>
<munderover>
<mo>∑</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>B</mi>
</munderover>
<msub>
<mi>b</mi>
<mi>i</mi>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">\displaystyle\alpha = \sum_{i=1}^A a_i, \beta = \sum_{i=1}^B b_i
</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span
class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace"
style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut"
style="height:3.106em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span
class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span
style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span
class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span
class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span
class="pstrut" style="height:3.05em;"></span><span><span
class="mop op-symbol large-op">∑</span></span></span><span
style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span
class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">A</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace"
style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span
class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"
style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span
class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">i</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span
class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span
class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace"
style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace"
style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut"
style="height:3.106em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span
class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span
style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span
class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span
class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span
class="pstrut" style="height:3.05em;"></span><span><span
class="mop op-symbol large-op">∑</span></span></span><span
style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight"
style="margin-right:0.05017em;">B</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace"
style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span
class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"
style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span
class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">i</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 。</p>
<p>将 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>α</mi>
</mrow>
<annotation encoding="application/x-tex">\alpha</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.4306em;"></span><span class="mord mathnormal"
style="margin-right:0.0037em;">α</span></span></span></span> 和 <span class="katex"><span
class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>β</mi>
</mrow>
<annotation encoding="application/x-tex">\beta</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal"
style="margin-right:0.05278em;">β</span></span></span></span> 皆均分成 <span class="katex"><span
class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>N</mi>
</mrow>
<annotation encoding="application/x-tex">N</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.6833em;"></span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span></span> 个等份，然后将其交错均匀排列，形成一个虚拟的交易序列，形如：</p>
<blockquote>
<p>1. 投入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>α</mi>
<mi mathvariant="normal">/</mi>
<mi>N</mi>
</mrow>
<annotation encoding="application/x-tex">\alpha/N</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal"
style="margin-right:0.0037em;">α</span><span class="mord">/</span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span></span> 个代币 X ；</p>
<p>2. 投入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>α</mi>
<mi mathvariant="normal">/</mi>
<mi>N</mi>
</mrow>
<annotation encoding="application/x-tex">\alpha/N</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal"
style="margin-right:0.0037em;">α</span><span class="mord">/</span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span></span> 个代币 Y ；</p>
<p>3. 投入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>α</mi>
<mi mathvariant="normal">/</mi>
<mi>N</mi>
</mrow>
<annotation encoding="application/x-tex">\alpha/N</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal"
style="margin-right:0.0037em;">α</span><span class="mord">/</span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span></span> 个代币 X ；</p>
<p>4. 投入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>α</mi>
<mi mathvariant="normal">/</mi>
<mi>N</mi>
</mrow>
<annotation encoding="application/x-tex">\alpha/N</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal"
style="margin-right:0.0037em;">α</span><span class="mord">/</span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span></span> 个代币 Y ；</p>
<p>......</p>
<p>2n-1. 投入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>α</mi>
<mi mathvariant="normal">/</mi>
<mi>N</mi>
</mrow>
<annotation encoding="application/x-tex">\alpha/N</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal"
style="margin-right:0.0037em;">α</span><span class="mord">/</span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span></span> 个代币 X ；</p>
<p>2n. 投入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>α</mi>
<mi mathvariant="normal">/</mi>
<mi>N</mi>
</mrow>
<annotation encoding="application/x-tex">\alpha/N</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal"
style="margin-right:0.0037em;">α</span><span class="mord">/</span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span></span> 个代币 Y ；</p>
</blockquote>
<p>在第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>i</mi>
</mrow>
<annotation encoding="application/x-tex">i</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 笔交易后，流动性池中的
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<msub>
<mi>x</mi>
<mi>i</mi>
</msub>
<mo separator="true">,</mo>
<msub>
<mi>y</mi>
<mi>i</mi>
</msub>
</mrow>
<annotation encoding="application/x-tex">x_i,y_i</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span
class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">i</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span
class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span
class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span
class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">i</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为</p>
<ul>
<li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>i</mi>
</mrow>
<annotation encoding="application/x-tex">i</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 为奇数， <span
class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle scriptlevel="0" displaystyle="true">
<msub>
<mi>x</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow>
<mi>i</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mfrac>
<mi>α</mi>
<mi>N</mi>
</mfrac>
<mo separator="true">,</mo>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mi>k</mi>
<mrow>
<msub>
<mi>x</mi>
<mrow>
<mi>i</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mfrac>
<mi>α</mi>
<mi>N</mi>
</mfrac>
</mrow>
</mfrac>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">\displaystyle x_i = x_{i-1} + \frac{\alpha}{N}, y =
\frac{k}{x_{i-1} + \frac{\alpha}{N}}</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span
class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">i</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace"
style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace"
style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut"
style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span
class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight"><span class="mord mathnormal mtight">i</span><span
class="mbin mtight">−</span><span
class="mord mtight">1</span></span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace"
style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace"
style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut"
style="height:1.7936em;vertical-align:-0.686em;"></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:1.1076em;"><span style="top:-2.314em;"><span
class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span><span style="top:-3.23em;"><span
class="pstrut" style="height:3em;"></span><span class="frac-line"
style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span
class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.0037em;">α</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.686em;"><span></span></span></span></span></span><span
class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace"
style="margin-right:0.1667em;"></span><span class="mord mathnormal"
style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span
class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span
class="base"><span class="strut" style="height:2.4024em;vertical-align:-1.031em;"></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:1.3714em;"><span
style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span
class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span
class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"
style="height:0.3117em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight"><span class="mord mathnormal mtight">i</span><span
class="mbin mtight">−</span><span
class="mord mtight">1</span></span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.2083em;"><span></span></span></span></span></span></span><span
class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span
class="mspace" style="margin-right:0.2222em;"></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:0.6954em;"><span style="top:-2.655em;"><span class="pstrut"
style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight"><span class="mord mathnormal mtight"
style="margin-right:0.10903em;">N</span></span></span></span><span
style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span
class="frac-line" style="border-bottom-width:0.04em;"></span></span><span
style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span
class="mord mathnormal mtight"
style="margin-right:0.0037em;">α</span></span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.345em;"><span></span></span></span></span></span><span
class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.23em;"><span
class="pstrut" style="height:3em;"></span><span class="frac-line"
style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span
class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.03148em;">k</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:1.031em;"><span></span></span></span></span></span><span
class="mclose nulldelimiter"></span></span></span></span></span></li>
<li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>i</mi>
</mrow>
<annotation encoding="application/x-tex">i</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 为偶数， <span
class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle scriptlevel="0" displaystyle="true">
<msub>
<mi>x</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<mfrac>
<mi>k</mi>
<mrow>
<msub>
<mi>y</mi>
<mrow>
<mi>i</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mfrac>
<mi>β</mi>
<mi>N</mi>
</mfrac>
</mrow>
</mfrac>
<mo separator="true">,</mo>
<msub>
<mi>y</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<msub>
<mi>y</mi>
<mrow>
<mi>i</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mfrac>
<mi>β</mi>
<mi>N</mi>
</mfrac>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">\displaystyle x_i = \frac{k}{y_{i-1} + \frac{\beta}{N}}, y_i =
y_{i-1} + \frac{\beta}{N}</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span
class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">i</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace"
style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace"
style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut"
style="height:2.5387em;vertical-align:-1.1672em;"></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:1.3714em;"><span style="top:-2.1778em;"><span
class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span
class="mord mathnormal" style="margin-right:0.03588em;">y</span><span
class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"
style="height:0.3117em;"><span
style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span
class="pstrut" style="height:2.7em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span
class="mord mathnormal mtight">i</span><span
class="mbin mtight">−</span><span
class="mord mtight">1</span></span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.2083em;"><span></span></span></span></span></span></span><span
class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span
class="mspace" style="margin-right:0.2222em;"></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:0.9322em;"><span style="top:-2.655em;"><span class="pstrut"
style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight"><span class="mord mathnormal mtight"
style="margin-right:0.10903em;">N</span></span></span></span><span
style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span
class="frac-line" style="border-bottom-width:0.04em;"></span></span><span
style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span
class="mord mathnormal mtight"
style="margin-right:0.05278em;">β</span></span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.345em;"><span></span></span></span></span></span><span
class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.23em;"><span
class="pstrut" style="height:3em;"></span><span class="frac-line"
style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span
class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.03148em;">k</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:1.1672em;"><span></span></span></span></span></span><span
class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace"
style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">i</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace"
style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace"
style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut"
style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight"><span class="mord mathnormal mtight">i</span><span
class="mbin mtight">−</span><span
class="mord mtight">1</span></span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace"
style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace"
style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut"
style="height:2.0574em;vertical-align:-0.686em;"></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:1.3714em;"><span style="top:-2.314em;"><span
class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span><span style="top:-3.23em;"><span
class="pstrut" style="height:3em;"></span><span class="frac-line"
style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span
class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.05278em;">β</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.686em;"><span></span></span></span></span></span><span
class="mclose nulldelimiter"></span></span></span></span></span></li>
</ul>
<p>随着 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>N</mi>
</mrow>
<annotation encoding="application/x-tex">N</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.6833em;"></span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span></span></span></span> 的增加， <span class="katex"><span
class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle scriptlevel="0" displaystyle="true">
<munder>
<mrow>
<mi>lim</mi>
<mo></mo>
</mrow>
<mrow>
<mi>N</mi>
<mo>→</mo>
<mi mathvariant="normal">∞</mi>
</mrow>
</munder>
<msub>
<mi>x</mi>
<mrow>
<mn>2</mn>
<mi>N</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">\displaystyle\lim_{N\to\infty} x_{2N}</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:1.4388em;vertical-align:-0.7443em;"></span><span class="mop op-limits"><span
class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span
style="top:-2.3557em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span
class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span><span
class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span
style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span
class="mop">lim</span></span></span></span><span class="vlist-s"></span></span><span
class="vlist-r"><span class="vlist"
style="height:0.7443em;"><span></span></span></span></span></span><span class="mspace"
style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span
class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"
style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span
class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight"
style="margin-right:0.10903em;">N</span></span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>
必然存在，证明如下：</p>
<ol>
<li>当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>i</mi>
</mrow>
<annotation encoding="application/x-tex">i</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 为偶数时，易得 <span
class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle scriptlevel="0" displaystyle="true">
<msub>
<mi>x</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mi>N</mi>
<msub>
<mi>x</mi>
<msub>
<mi>i</mi>
<mn>2</mn>
</msub>
</msub>
<mo>+</mo>
<mi>α</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>k</mi>
<mi>N</mi>
<mo>+</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow>
<mi>i</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<mfrac>
<mi>α</mi>
<mi>N</mi>
</mfrac>
<mo stretchy="false">)</mo>
<mi>β</mi>
</mrow>
</mfrac>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">\displaystyle x_i = \frac{kNx_{i_2}+\alpha
k}{kN+(x_{i-2}+\frac{\alpha}{N})\beta}</annotation>
</semantics>
</math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut"
style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span
class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mathnormal mtight">i</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace"
style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace"
style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut"
style="height:2.4024em;vertical-align:-1.031em;"></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:1.3714em;"><span style="top:-2.314em;"><span
class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.03148em;">k</span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span><span class="mspace"
style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace"
style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span
class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight"><span class="mord mathnormal mtight">i</span><span
class="mbin mtight">−</span><span
class="mord mtight">2</span></span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.2083em;"><span></span></span></span></span></span></span><span
class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span
class="mspace" style="margin-right:0.2222em;"></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:0.6954em;"><span style="top:-2.655em;"><span class="pstrut"
style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight"><span class="mord mathnormal mtight"
style="margin-right:0.10903em;">N</span></span></span></span><span
style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span
class="frac-line" style="border-bottom-width:0.04em;"></span></span><span
style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span
class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span
class="mord mathnormal mtight"
style="margin-right:0.0037em;">α</span></span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span class="vlist"
style="height:0.345em;"><span></span></span></span></span></span><span
class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span
class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span
style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line"
style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span
class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal"
style="margin-right:0.03148em;">k</span><span class="mord mathnormal"
style="margin-right:0.10903em;">N</span><span class="mord"><span
class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span
class="vlist-r"><span class="vlist" style="height:0.3117em;"><span
style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut"
style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span
class="mord mtight"><span class="mord mtight"><span
class="mord mathnormal mtight">i</span><span class="msupsub"><span
class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"
style="height:0.3173em;"><span
style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span
class="pstrut" style="height:2.5em;"></span><span
class="sizing reset-size3 size1 mtight"><span
class="mord mtight">2</span></span></span></span><span
class="vlist-s"></span></span><span class="vlist-r"><span
class="vlist"