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</div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-沈括隙积术" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#沈括隙积术"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>沈括隙积术</span> </div> </a> <ul id="toc-沈括隙积术-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-杨辉垛积术" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#杨辉垛积术"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>杨辉垛积术</span> </div> </a> <button aria-controls="toc-杨辉垛积术-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关杨辉垛积术子章节</span> </button> <ul id="toc-杨辉垛积术-sublist" class="vector-toc-list"> <li id="toc-方垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#方垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>方垛</span> </div> </a> <ul id="toc-方垛-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-刍童垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#刍童垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>刍童垛</span> </div> </a> <ul id="toc-刍童垛-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-三角垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#三角垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>三角垛</span> </div> </a> <ul id="toc-三角垛-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-朱世杰垛积术" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#朱世杰垛积术"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>朱世杰垛积术</span> </div> </a> <button aria-controls="toc-朱世杰垛积术-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关朱世杰垛积术子章节</span> </button> <ul id="toc-朱世杰垛积术-sublist" class="vector-toc-list"> <li id="toc-三角垛_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#三角垛_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>三角垛</span> </div> </a> <ul id="toc-三角垛_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-三角落一形垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#三角落一形垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>三角落一形垛</span> </div> </a> <ul id="toc-三角落一形垛-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-四角落一形垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#四角落一形垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>四角落一形垛</span> </div> </a> <ul id="toc-四角落一形垛-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-岚峰形垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#岚峰形垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>岚峰形垛</span> </div> </a> <ul id="toc-岚峰形垛-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-三角岚峰形垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#三角岚峰形垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>三角岚峰形垛</span> </div> </a> <ul id="toc-三角岚峰形垛-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-撒星更落一形垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#撒星更落一形垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>撒星更落一形垛</span> </div> </a> <ul id="toc-撒星更落一形垛-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-三角撒星更落一形垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#三角撒星更落一形垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>三角撒星更落一形垛</span> </div> </a> <ul id="toc-三角撒星更落一形垛-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-四角岚峰形垛" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#四角岚峰形垛"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>四角岚峰形垛</span> </div> </a> <ul id="toc-四角岚峰形垛-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" 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class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><p><b>垛积术</b>，也称<b>隙积术</b>，实质上是一种<a href="/wiki/%E9%AB%98%E9%98%B6%E7%AD%89%E5%B7%AE%E7%BA%A7%E6%95%B0%E6%B1%82%E5%92%8C" class="mw-redirect" title="高阶等差级数求和">高阶等差级数求和</a>问题。由北宋<a href="/wiki/%E6%B2%88%E6%8B%AC" title="沈括">沈括</a>首開先河，南宋<a href="/wiki/%E6%9D%A8%E8%BE%89" title="杨辉">杨辉</a>和元朝<a href="/wiki/%E6%9C%B1%E4%B8%96%E6%9D%B0" class="mw-redirect" title="朱世杰">朱世杰</a>多有贡献。 </p><p>沈括在《<a href="/wiki/%E6%A2%A6%E6%BA%AA%E7%AC%94%E8%B0%88" title="梦溪笔谈">梦溪笔谈</a>》卷十八《技艺》篇首创隙积术，是用來研究某種物品按規律堆積起來求其總數問題。隙积是指酒甕之类的物品，往上堆積成台形之狀，求其總數，這是二階等差級數求和問題。至於垛積是堆垛求積的意思。垛积术是杨辉繼沈括的隙積術之後，開創高阶等差级数的研究。元代朱世杰則將垛积术的研究推向最高峰，他使用的<a href="/wiki/%E6%8B%9B%E5%B7%AE%E6%9C%AF" title="招差术">招差術</a>实际上是解决了任意高阶等差级数的有限项求和问题。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="沈括隙积术"><span id=".E6.B2.88.E6.8B.AC.E9.9A.99.E7.A7.AF.E6.9C.AF"></span>沈括隙积术</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=1" title="编辑章节：沈括隙积术"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>北宋<a href="/wiki/%E6%B2%88%E6%8B%AC" title="沈括">沈括</a>在《<a href="/wiki/%E6%A2%A6%E6%BA%AA%E7%AC%94%E8%B0%88" title="梦溪笔谈">梦溪笔谈</a>》卷十八《技艺》篇，首创隙积术：<i>隙積者，謂積之有隙者，如累棋、層壇及灑家積罌之類。雖似覆鬥，四面皆殺，緣有刻缺及虛隙之處，用芻童法求之，常失於數少。餘思而得之，用爭童法為上位；下位別列：下廣以上廣減之，余者以高乘之，六而一，並入上位。假令積罌：最上行縱橫各二罌，最下行各十二罌，行行相次。先以上二行相次，率至十二，當十一行也。以芻童法求之，倍上行長得四，並入下長得十六，以上廣乘之，得之三十二；又倍下行長得二十四，並入上長，得二十六，以下廣乘之，得三百一十二；並二位得三百四十四，以高乘之，得三千七百八十四。重列下廣十二，以上廣減之，餘十，以高乘之，得一百一十，並入上位，得三千八百九十四；六而一，得六百四十九，此為罌數也。芻童求見實方之積，隙積求見合角不盡，益出羨積也</i> </p><p>一个层罈，共<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>层，上面<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\times b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b420244850c1a22be4c326f91e146db8b037f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.068ex; height:2.176ex;" alt="{\displaystyle a\times b}"></span>，下底<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\times d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>×</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\times d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c082db4221f29ebe364ff52ee23d061100eeeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle c\times d}"></span>， </p><p>这是二阶等差级数求和问题： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab+(a+1)(b+1)+(a+2)(b+2)+...+(a+h-1)(b+h-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>h</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab+(a+1)(b+1)+(a+2)(b+2)+...+(a+h-1)(b+h-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46cf142f52a7a5cf1cd2eceb38017f9c01fdd0c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.54ex; height:2.843ex;" alt="{\displaystyle ab+(a+1)(b+1)+(a+2)(b+2)+...+(a+h-1)(b+h-1)}"></span></dd></dl> <p>沈括给出的公式 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {h}{6}}((2a+c)b+(2c+a)d+(d-b))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mn>6</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {h}{6}}((2a+c)b+(2c+a)d+(d-b))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/791a6f83ec7d5d6c03350f6a5c9357d9ac3841ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.839ex; height:5.343ex;" alt="{\displaystyle {\frac {h}{6}}((2a+c)b+(2c+a)d+(d-b))}"></span><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="杨辉垛积术"><span id=".E6.9D.A8.E8.BE.89.E5.9E.9B.E7.A7.AF.E6.9C.AF"></span>杨辉垛积术</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=2" title="编辑章节：杨辉垛积术"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>杨辉在《详解九章算法》《商功》篇阐述了方垛，刍甍垛，刍童垛，和三角垛。 </p> <div class="mw-heading mw-heading3"><h3 id="方垛"><span id=".E6.96.B9.E5.9E.9B"></span>方垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=3" title="编辑章节：方垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>果子以垛，下方十四个，问计几何？</i> <i>术曰：下方加一，乘下方为平积。又加半为高，以乘下方为高积。如三而一</i>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+4+9+16....n^{2}={\frac {1}{3}}n(n+1)(n+{\frac {1}{2}})={\frac {1}{6}}n(n+1)(2n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>9</mn> <mo>+</mo> <mn>16....</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+4+9+16....n^{2}={\frac {1}{3}}n(n+1)(n+{\frac {1}{2}})={\frac {1}{6}}n(n+1)(2n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927a9ed1da9f929ab097127c0a2ee7dac92cb4b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.179ex; height:5.176ex;" alt="{\displaystyle 1+4+9+16....n^{2}={\frac {1}{3}}n(n+1)(n+{\frac {1}{2}})={\frac {1}{6}}n(n+1)(2n+1)}"></span>。<sup id="cite_ref-#1_2-0" class="reference"><a href="#cite_note-#1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="刍童垛"><span id=".E5.88.8D.E7.AB.A5.E5.9E.9B"></span>刍童垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=4" title="编辑章节：刍童垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>即长方形立体垛,上面长<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>个,宽<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>个,高<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>个: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4*2+5*3+6*4+7*5+\dots +(n+h-1)*(m+h-1)={\frac {h}{6}}((2n+n+h-1))*m+(2(n+h-1)+n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>∗</mo> <mn>2</mn> <mo>+</mo> <mn>5</mn> <mo>∗</mo> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mo>∗</mo> <mn>4</mn> <mo>+</mo> <mn>7</mn> <mo>∗</mo> <mn>5</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>h</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>h</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mn>6</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mi>h</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>m</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>h</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4*2+5*3+6*4+7*5+\dots +(n+h-1)*(m+h-1)={\frac {h}{6}}((2n+n+h-1))*m+(2(n+h-1)+n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7e0fd0e361b6f5e211f0c94ec31ce30325bb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:110.152ex; height:5.343ex;" alt="{\displaystyle 4*2+5*3+6*4+7*5+\dots +(n+h-1)*(m+h-1)={\frac {h}{6}}((2n+n+h-1))*m+(2(n+h-1)+n)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="三角垛"><span id=".E4.B8.89.E8.A7.92.E5.9E.9B"></span>三角垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=5" title="编辑章节：三角垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>三角垛下广一面十二个，上尖，高十二个，问：计几何?</i> </p> <dl><dd>术曰：下广加一，乘下广。平积，下广加二乘之，立高方积，如六而一。</dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+3+6+10+\dots {\frac {n(n+1)}{2}}={\frac {1}{6}}n(n+1)(n+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mo>+</mo> <mn>10</mn> <mo>+</mo> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+3+6+10+\dots {\frac {n(n+1)}{2}}={\frac {1}{6}}n(n+1)(n+2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d94c1a7d24254baa1ce9fe128960ad3a38bbbe88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.627ex; height:5.676ex;" alt="{\displaystyle 1+3+6+10+\dots {\frac {n(n+1)}{2}}={\frac {1}{6}}n(n+1)(n+2)}"></span><sup id="cite_ref-#1_2-1" class="reference"><a href="#cite_note-#1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="朱世杰垛积术"><span id=".E6.9C.B1.E4.B8.96.E6.9D.B0.E5.9E.9B.E7.A7.AF.E6.9C.AF"></span>朱世杰垛积术</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=6" title="编辑章节：朱世杰垛积术"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="三角垛_2"><span id=".E4.B8.89.E8.A7.92.E5.9E.9B_2"></span>三角垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=7" title="编辑章节：三角垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>《<a href="/wiki/%E5%9B%9B%E5%85%83%E7%8E%89%E9%89%B4" title="四元玉鉴">四元玉鉴</a>》《果垛叠藏》第一问： </p><p>“今有三角垛果子一所，值钱一贯三百二十文，只云从上一个值钱二文，次下层层每个累贵一文，问底子每面几何？” </p> <dl><dd>答曰：九个。</dd></dl> <p>术曰：立天元一为每个底子，如积求之，得三万一千六百八十为益实十为从方，二十一为从上廉，一十四为下廉，三为从隅，三桀方开之，得每个底子，合问。 </p><p>三角垛级数 </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+3+6+10+...+{\frac {1}{2}}n(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mo>+</mo> <mn>10</mn> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+3+6+10+...+{\frac {1}{2}}n(n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2be318838069dede4d2553f3102b4163e9b1d76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.652ex; height:5.176ex;" alt="{\displaystyle 1+3+6+10+...+{\frac {1}{2}}n(n+1)}"></span> </p><p>三角垛自上而下，每边的果子数是： </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,2,3,4,5,6,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,3,4,5,6,\dots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5785e3a078b8f66688b2f474ae26a7dbc7cf8d67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.717ex; height:2.509ex;" alt="{\displaystyle 1,2,3,4,5,6,\dots ,n}"></span> </p><p>自上而下，每个果子值钱： </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,3,4,5,6,7,\dots ,(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,3,4,5,6,7,\dots ,(n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d44bf391b32fa7e284589543b5530957abb3a34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.53ex; height:2.843ex;" alt="{\displaystyle 2,3,4,5,6,7,\dots ,(n+1)}"></span> </p><p>三角果子垛价值V由下列级数表示 </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=2+9+24+50+90+147+224+\dots +{\frac {1}{2}}n(n+1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>9</mn> <mo>+</mo> <mn>24</mn> <mo>+</mo> <mn>50</mn> <mo>+</mo> <mn>90</mn> <mo>+</mo> <mn>147</mn> <mo>+</mo> <mn>224</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=2+9+24+50+90+147+224+\dots +{\frac {1}{2}}n(n+1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38278d2e431bee3dd53d8708c7be4ed7da57a679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.601ex; height:5.176ex;" alt="{\displaystyle v=2+9+24+50+90+147+224+\dots +{\frac {1}{2}}n(n+1)^{2}}"></span> </p><p>这是一个已知级数和，倒求 n 的数学问题。 </p><p>朱世杰用天元术，令天元一为每底边的果子数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c4d06047faa5d69e80c075e139a747ebbbaac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.823ex; height:1.676ex;" alt="{\displaystyle x=n}"></span> </p><p>朱世杰用的求和公式：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\frac {1}{2*3*4}}(3x+5)*x*(x+1)*(x+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>∗</mo> <mn>3</mn> <mo>∗</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>x</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {1}{2*3*4}}(3x+5)*x*(x+1)*(x+2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5320461211b7ca66e227346dccb8e658a10ebcc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.441ex; height:5.343ex;" alt="{\displaystyle v={\frac {1}{2*3*4}}(3x+5)*x*(x+1)*(x+2)}"></span> </p><p><br /> </p><p>今<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=1320}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>1320</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=1320}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca77e8d0f270d43bfe30f5df31fb25b0e95e829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.876ex; height:2.176ex;" alt="{\displaystyle v=1320}"></span>得 </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3*x^{4}+14x^{3}+21x^{2}+10x-31680=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>∗</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>14</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>21</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>10</mn> <mi>x</mi> <mo>−</mo> <mn>31680</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3*x^{4}+14x^{3}+21x^{2}+10x-31680=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e28af66c2c10f659556ec1a95175ff36b853f68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:40.248ex; height:2.843ex;" alt="{\displaystyle 3*x^{4}+14x^{3}+21x^{2}+10x-31680=0}"></span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>解之，得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=n=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>n</mi> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=n=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d80634ef74d5bbd8deee6a8ec8e4b0cea7d85ba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.084ex; height:2.176ex;" alt="{\displaystyle x=n=9}"></span>。 </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=2+9+24+50+90+147+224+324+450=1320}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>9</mn> <mo>+</mo> <mn>24</mn> <mo>+</mo> <mn>50</mn> <mo>+</mo> <mn>90</mn> <mo>+</mo> <mn>147</mn> <mo>+</mo> <mn>224</mn> <mo>+</mo> <mn>324</mn> <mo>+</mo> <mn>450</mn> <mo>=</mo> <mn>1320</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=2+9+24+50+90+147+224+324+450=1320}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51f97b2c84e4bc07e60ea95097ca34972dad3a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:57.947ex; height:2.343ex;" alt="{\displaystyle v=2+9+24+50+90+147+224+324+450=1320}"></span>。 </p> <div class="mw-heading mw-heading3"><h3 id="三角落一形垛"><span id=".E4.B8.89.E8.A7.92.E8.90.BD.E4.B8.80.E5.BD.A2.E5.9E.9B"></span>三角落一形垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=8" title="编辑章节：三角落一形垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1*2*3+2*3*4+3*4*5+\dots +n(n+1)(n+2)={\frac {1}{4}}n(n+1)(n+2)(n+3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>∗</mo> <mn>2</mn> <mo>∗</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>∗</mo> <mn>3</mn> <mo>∗</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> <mo>∗</mo> <mn>4</mn> <mo>∗</mo> <mn>5</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1*2*3+2*3*4+3*4*5+\dots +n(n+1)(n+2)={\frac {1}{4}}n(n+1)(n+2)(n+3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e71c133c9f1fee6a3531dcc654224d214e3671a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:81.636ex; height:5.176ex;" alt="{\displaystyle 1*2*3+2*3*4+3*4*5+\dots +n(n+1)(n+2)={\frac {1}{4}}n(n+1)(n+2)(n+3)}"></span><sup id="cite_ref-#2_4-0" class="reference"><a href="#cite_note-#2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="四角落一形垛"><span id=".E5.9B.9B.E8.A7.92.E8.90.BD.E4.B8.80.E5.BD.A2.E5.9E.9B"></span>四角落一形垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=9" title="编辑章节：四角落一形垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1*2*3+2*3*5+3*4*7+\dots +n(n+1)(2n+2)={\frac {1}{2}}n(n+1)^{2}(n+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>∗</mo> <mn>2</mn> <mo>∗</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>∗</mo> <mn>3</mn> <mo>∗</mo> <mn>5</mn> <mo>+</mo> <mn>3</mn> <mo>∗</mo> <mn>4</mn> <mo>∗</mo> <mn>7</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1*2*3+2*3*5+3*4*7+\dots +n(n+1)(2n+2)={\frac {1}{2}}n(n+1)^{2}(n+2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0c671ce428113327b3c72f23ae46bb42f0e6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:76.646ex; height:5.176ex;" alt="{\displaystyle 1*2*3+2*3*5+3*4*7+\dots +n(n+1)(2n+2)={\frac {1}{2}}n(n+1)^{2}(n+2)}"></span><sup id="cite_ref-#2_4-1" class="reference"><a href="#cite_note-#2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="岚峰形垛"><span id=".E5.B2.9A.E5.B3.B0.E5.BD.A2.E5.9E.9B"></span>岚峰形垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=10" title="编辑章节：岚峰形垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+6+18+\dots +n^{2}(n+1)={\frac {1}{24}}n(n+1)(n+2)(3n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>6</mn> <mo>+</mo> <mn>18</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+6+18+\dots +n^{2}(n+1)={\frac {1}{24}}n(n+1)(n+2)(3n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9399dcda9fb0e1539d12993bd9f52ad2d944b41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.828ex; height:5.176ex;" alt="{\displaystyle 1+6+18+\dots +n^{2}(n+1)={\frac {1}{24}}n(n+1)(n+2)(3n+1)}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="三角岚峰形垛"><span id=".E4.B8.89.E8.A7.92.E5.B2.9A.E5.B3.B0.E5.BD.A2.E5.9E.9B"></span>三角岚峰形垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=11" title="编辑章节：三角岚峰形垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6+48+180+\dots +n^{2}(n+1)(n+2)={\frac {1}{120}}n(n+1)(n+2)(n+3)(4n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>+</mo> <mn>48</mn> <mo>+</mo> <mn>180</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>120</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6+48+180+\dots +n^{2}(n+1)(n+2)={\frac {1}{120}}n(n+1)(n+2)(n+3)(4n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c096ec0b6474c87585587c2b2860531cc2c39c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:76.729ex; height:5.176ex;" alt="{\displaystyle 6+48+180+\dots +n^{2}(n+1)(n+2)={\frac {1}{120}}n(n+1)(n+2)(n+3)(4n+1)}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="撒星更落一形垛"><span id=".E6.92.92.E6.98.9F.E6.9B.B4.E8.90.BD.E4.B8.80.E5.BD.A2.E5.9E.9B"></span>撒星更落一形垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=12" title="编辑章节：撒星更落一形垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+5+15+35+70+\dots +{\frac {1}{24}}n(n+1)(n+2)(n+3)={\frac {1}{120}}n(n+1)(n+2)(n+3)(n+4)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>5</mn> <mo>+</mo> <mn>15</mn> <mo>+</mo> <mn>35</mn> <mo>+</mo> <mn>70</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>120</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+5+15+35+70+\dots +{\frac {1}{24}}n(n+1)(n+2)(n+3)={\frac {1}{120}}n(n+1)(n+2)(n+3)(n+4)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295a9aedced7ce9ccaf4a277fb5ee44808ce4f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:92.886ex; height:5.176ex;" alt="{\displaystyle 1+5+15+35+70+\dots +{\frac {1}{24}}n(n+1)(n+2)(n+3)={\frac {1}{120}}n(n+1)(n+2)(n+3)(n+4)}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="三角撒星更落一形垛"><span id=".E4.B8.89.E8.A7.92.E6.92.92.E6.98.9F.E6.9B.B4.E8.90.BD.E4.B8.80.E5.BD.A2.E5.9E.9B"></span>三角撒星更落一形垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=13" title="编辑章节：三角撒星更落一形垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+6+21+56+126+\dots +{\frac {1}{120}}n(n+1)(n+2)(n+3)(n+4)={\frac {1}{720}}n(n+1)(n+2)(n+3)(n+4)(n+5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>6</mn> <mo>+</mo> <mn>21</mn> <mo>+</mo> <mn>56</mn> <mo>+</mo> <mn>126</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>120</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>720</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+6+21+56+126+\dots +{\frac {1}{120}}n(n+1)(n+2)(n+3)(n+4)={\frac {1}{720}}n(n+1)(n+2)(n+3)(n+4)(n+5)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f7c7ce2bc561e3305edf594a94fc177559e8a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:109.624ex; height:5.343ex;" alt="{\displaystyle 1+6+21+56+126+\dots +{\frac {1}{120}}n(n+1)(n+2)(n+3)(n+4)={\frac {1}{720}}n(n+1)(n+2)(n+3)(n+4)(n+5)}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="四角岚峰形垛"><span id=".E5.9B.9B.E8.A7.92.E5.B2.9A.E5.B3.B0.E5.BD.A2.E5.9E.9B"></span>四角岚峰形垛</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=14" title="编辑章节：四角岚峰形垛"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6+90+336+900+\dots +n^{2}(n+1)(2n+1)={\frac {1}{10}}n(n+1)(n+2)(n(4n+1+{\frac {1}{2}})+(4n+{\frac {1}{2}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>+</mo> <mn>90</mn> <mo>+</mo> <mn>336</mn> <mo>+</mo> <mn>900</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6+90+336+900+\dots +n^{2}(n+1)(2n+1)={\frac {1}{10}}n(n+1)(n+2)(n(4n+1+{\frac {1}{2}})+(4n+{\frac {1}{2}}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddcf7169f6a5cadc7c5d80ba5942537abef29a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:95.939ex; height:5.176ex;" alt="{\displaystyle 6+90+336+900+\dots +n^{2}(n+1)(2n+1)={\frac {1}{10}}n(n+1)(n+2)(n(4n+1+{\frac {1}{2}})+(4n+{\frac {1}{2}}))}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%9A%99%E7%A9%8D%E8%A1%93&action=edit&section=15" title="编辑章节：参考文献"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">吴文俊主编《<a href="/wiki/%E4%B8%AD%E5%9B%BD%E6%95%B0%E5%AD%A6%E5%8F%B2%E5%A4%A7%E7%B3%BB" class="mw-disambig" title="中国数学史大系">中国数学史大系</a>》第五卷北宋沈括 73页 <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7303049266" class="internal mw-magiclink-isbn">ISBN 7-303-04926-6</a>/O</span> </li> <li id="cite_note-#1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-#1_2-0"><sup><b>2.0</b></sup></a> <a href="#cite_ref-#1_2-1"><sup><b>2.1</b></sup></a></span> <span class="reference-text">吴文俊主编《中国数学史大系》第五卷南宋杨辉 592页</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">万有文库第二集朱世杰撰罗士琳草（中）卷下之一六四六-六四八</span> </li> <li id="cite_note-#2-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-#2_4-0"><sup><b>4.0</b></sup></a> <a href="#cite_ref-#2_4-1"><sup><b>4.1</b></sup></a></span> <span class="reference-text">《李俨.钱宝琮科学史全集》卷3 294页辽宁教育出版社. 1998</span> </li> </ol></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:" :"}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · 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