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<title>Triangular number</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Triangular number</span></span>

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<p>A <b>triangular number</b> or <b>triangle number</b> counts objects arranged in an <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>. Triangular numbers are a type of <a href="Figurate_number" title="Figurate number">figurate number</a>, other examples being <a href="Square_number" title="Square number">square numbers</a> and <a href="Cube_(algebra)#In_integers" title="Cube (algebra)">cube numbers</a>. The <span class="texhtml mvar" style="font-style:italic;">n</span>th triangular number is the number of dots in the triangular arrangement with <span class="texhtml mvar" style="font-style:italic;">n</span> dots on each side, and is equal to the sum of the <span class="texhtml mvar" style="font-style:italic;">n</span> <a href="Natural_number" title="Natural number">natural numbers</a> from 1 to <span class="texhtml mvar" style="font-style:italic;">n</span>. The first 100 terms <a href="Integer_sequence" title="Integer sequence">sequence</a> of triangular numbers, starting with the <a href="Empty_sum" title="Empty sum">0th triangular number</a>, are

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</style><div class="block-indent" style="padding-left: 1.6em;">0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1830, 1891, 1953, 2016, 2080, 2145, 2211, 2278, 2346, 2415, 2485, 2556, 2628, 2701, 2775, 2850, 2926, 3003, 3081, 3160, 3240, 3321, 3403, 3486, 3570, 3655, 3741, 3828, 3916, 4005, 4095, 4186, 4278, 4371, 4465, 4560, 4656, 4753, 4851, 4950, 5050...</div>

<p>(sequence <span class="nowrap external"><a href="https://oeis.org/A000217" class="extiw external" title="oeis:A000217">A000217</a></span> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)

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<div class="mw-heading mw-heading2"><h2 id="Formula">Formula</h2></div>

<p>The triangular numbers are given by the following explicit formulas:

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {\begin{aligned}T_{n}&=\sum _{k=1}^{n}k=1+2+\dotsb +n\\&={\frac {n^{2}+n{\vphantom {(n+1)}}}{2}}={\frac {n(n+1)}{2}}\\&={n+1 \choose 2}\end{aligned}}}</annotation>

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</math></span><img src="./da2be4366fef3ae13fdf9c172bd3bceb801b38f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:29.442ex; height:19.176ex;" alt="{\displaystyle \displaystyle {\begin{aligned}T_{n}&=\sum _{k=1}^{n}k=1+2+\dotsb +n\\&={\frac {n^{2}+n{\vphantom {(n+1)}}}{2}}={\frac {n(n+1)}{2}}\\&={n+1 \choose 2}\end{aligned}}}" loading="lazy"></span></div>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {n+1 \choose 2}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {n+1 \choose 2}}</annotation>

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</math></span><img src="./b304adfa8d11ffd7c6a7f9ebe10891b8db5eb611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.217ex; height:3.343ex;" alt="{\displaystyle \textstyle {n+1 \choose 2}}" loading="lazy"></span> is notation for a <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>. It represents the number of distinct pairs that can be selected from <span class="texhtml"><i>n</i> + 1</span> objects, and it is read aloud as "<span class="texhtml mvar" style="font-style:italic;">n</span> plus one choose two".

</p><p>The fact that the <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"></span>th triangular number equals <span class="mwe-math-element mwe-math-element-inline"><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(n+1)/2}">

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<annotation encoding="application/x-tex">{\displaystyle n(n+1)/2}</annotation>

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</math></span><img src="./880739d44f2dc75cf0c9404a2d4b984f1fbc610a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.926ex; height:2.843ex;" alt="{\displaystyle n(n+1)/2}" loading="lazy"></span> can be illustrated using a <a href="Proof_without_words" title="Proof without words">visual proof</a>.<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> For every triangular number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation>

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</math></span><img src="./4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy"></span>, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions <span class="mwe-math-element mwe-math-element-inline"><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\times (n+1)}">

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<annotation encoding="application/x-tex">{\displaystyle n\times (n+1)}</annotation>

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</math></span><img src="./b9a4c74010769d5fa9888e3f73f270126e963f0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.442ex; height:2.843ex;" alt="{\displaystyle n\times (n+1)}" loading="lazy"></span>, which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}={\frac {n(n+1)}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{n}={\frac {n(n+1)}{2}}}</annotation>

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</math></span><img src="./c38b21816ac2c7ffc218389b16b4de8ebf8b927d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.112ex; height:5.676ex;" alt="{\displaystyle T_{n}={\frac {n(n+1)}{2}}}" loading="lazy"></span>. The example <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{4}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{4}}</annotation>

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</math></span><img src="./83a8e4649fd334cf176ee7badff8eb949ecd3670.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{4}}" loading="lazy"></span> follows:

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<annotation encoding="application/x-tex">{\displaystyle 2T_{4}=4(4+1)=20}</annotation>

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</math></span><img src="./9c5e777e82a762d3e826e4b411a0d2f4e4cb6339.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.233ex; height:2.843ex;" alt="{\displaystyle 2T_{4}=4(4+1)=20}" loading="lazy"></span> (green plus yellow) implies that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{4}={\frac {4(4+1)}{2}}=10}">

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<annotation encoding="application/x-tex">{\displaystyle T_{4}={\frac {4(4+1)}{2}}=10}</annotation>

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</math></span><img src="./992da298290c9ba3d3e16562eaabde42b3335262.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.907ex; height:5.676ex;" alt="{\displaystyle T_{4}={\frac {4(4+1)}{2}}=10}" loading="lazy"></span> (green).    <span class="mw-default-size" typeof="mw:File"></span>

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<p>This formula can be proven formally using <a href="Mathematical_induction" title="Mathematical induction">mathematical induction</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It is clearly true for <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{1}=\sum _{k=1}^{1}k={\frac {1(1+1)}{2}}={\frac {2}{2}}=1.}">

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<annotation encoding="application/x-tex">{\displaystyle T_{1}=\sum _{k=1}^{1}k={\frac {1(1+1)}{2}}={\frac {2}{2}}=1.}</annotation>

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</p><p>Now assume that, for some natural number <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" 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}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{m}=\sum _{k=1}^{m}k={\frac {m(m+1)}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{m}=\sum _{k=1}^{m}k={\frac {m(m+1)}{2}}}</annotation>

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</math></span><img src="./cfe8ffc0ec539a0b04eb8cef331a2b0c0fdfec5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.912ex; height:6.843ex;" alt="{\displaystyle T_{m}=\sum _{k=1}^{m}k={\frac {m(m+1)}{2}}}" loading="lazy"></span>. We can then verify it for <span class="mwe-math-element mwe-math-element-inline"><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+1}">

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<annotation encoding="application/x-tex">{\displaystyle m+1}</annotation>

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</math></span><img src="./c6f7ed29a2b4a62d3b6af05cd91a58ffc6094201.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m+1}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{k=1}^{m+1}k&=\sum _{k=1}^{m}k+(m+1)\\&={\frac {m(m+1)}{2}}+m+1\\&={\frac {m^{2}+m}{2}}+{\frac {2m+2}{2}}\\&={\frac {m^{2}+3m+2}{2}}\\&={\frac {(m+1)(m+2)}{2}},\end{aligned}}}</annotation>

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</p><p>so if the formula is true for <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" 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}" loading="lazy"></span>, it is true for <span class="mwe-math-element mwe-math-element-inline"><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+1}">

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<annotation encoding="application/x-tex">{\displaystyle m+1}</annotation>

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</math></span><img src="./c6f7ed29a2b4a62d3b6af05cd91a58ffc6094201.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m+1}" loading="lazy"></span>. Since it is clearly true for <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>, it is therefore true for <span class="mwe-math-element mwe-math-element-inline"><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}">

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</math></span><img src="./991e33c6e207b12546f15bdfee8b5726eafbbb2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" loading="lazy"></span>, and ultimately all natural numbers <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"></span> by induction.

</p><p>The German mathematician and scientist, <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>, is said to have found this relationship in his early youth, by multiplying <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> pairs of numbers in the sum by the values of each pair <span class="texhtml"><i>n</i> + 1</span>.<sup id="cite_ref-Gauss_3-0" class="reference"><a href="#cite_note-Gauss-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the <a href="Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagoreans</a> in the 5th century BC.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The two formulas were described by the Irish monk <a href="Dicuil" title="Dicuil">Dicuil</a> in about 816 in his <a href="Computus" class="mw-redirect" title="Computus">Computus</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> An English translation of Dicuil's account is available.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>Occasionally it is necessary to compute large triangular numbers where the standard formula <code>t = n*(n+1)/2</code> would suffer <a href="Integer_overflow" title="Integer overflow">integer overflow</a> before the final division by 2. For example, <span class="texhtml"><i>T</i><sub>20</sub></span> = 210 < 256, so will fit into an <a href="8-bit_byte" class="mw-redirect" title="8-bit byte">8-bit byte</a>, but not the intermediate product 420. This can be solved by dividing either <span class="texhtml mvar" style="font-style:italic;">n</span> or <span class="texhtml mvar" style="font-style:italic;">n+1</span> by 2 before the multiplication, whichever is even. This does not require a <a href="Conditional_branch" class="mw-redirect" title="Conditional branch">conditional branch</a> if implemented as <code>t = (n|1) * ((n+1)/2)</code>. If <code>n</code> is odd, the <a href="Binary_OR" class="mw-redirect" title="Binary OR">binary OR</a> operation <code>n|1</code> has no effect, so this is equivalent to <code>t = n * ((n+1)/2)</code> and thus correct. If <code>n</code> is even, setting the low bit with <code>n|1</code> is the same as adding 1, while the 1 added before the division is <a href="Division_(mathematics)#Of_integers" title="Division (mathematics)">truncated away</a>, so this is equivalent to <code>t = (n+1) * (n/2)</code> and also correct.

</p>

<div class="mw-heading mw-heading2"><h2 id="Relations_to_other_figurate_numbers">Relations to other figurate numbers</h2></div>

<p>Triangular numbers have a wide variety of relations to other figurate numbers.

</p><p>Most simply, the sum of two consecutive triangular numbers is a square number, since:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

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<annotation encoding="application/x-tex">{\displaystyle T_{n-1}+T_{n}}</annotation>

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</math></span><img src="./99ef23205b1335ee73c54997463cf08c1d0de09b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle T_{n-1}+T_{n}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2}}\,n(n-1)+{\frac {1}{2}}\,n(n+1)}</annotation>

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</math></span><img src="./7e4fbf8a5f9617e2522175a843eb6596fe92b72e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.268ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{2}}\,n(n-1)+{\frac {1}{2}}\,n(n+1)}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2}}\,n{\Bigl (}(n-1)+(n+1){\Bigr )}}</annotation>

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</math></span><img src="./8179c8c3cc2c2063e228b653db986c3a45343884.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.263ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{2}}\,n{\Bigl (}(n-1)+(n+1){\Bigr )}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle =n^{2}}</annotation>

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</math></span><img src="./0bc02af512495c683b23922c2999e0cb49fd3b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.902ex; height:2.676ex;" alt="{\displaystyle =n^{2}}" loading="lazy"></span></dd></dl>

<p>with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum):

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<annotation encoding="application/x-tex">{\displaystyle T_{n}+T_{n-1}=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {\left(n-1\right)^{2}}{2}}+{\frac {n-1{\vphantom {\left(n-1\right)^{2}}}}{2}}\right)=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {n^{2}}{2}}-{\frac {n}{2}}\right)=n^{2}=(T_{n}-T_{n-1})^{2}.}</annotation>

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</p><p>This property, colloquially known as the theorem of <a href="Theon_of_Smyrna" title="Theon of Smyrna">Theon of Smyrna</a>,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> is visually demonstrated in the following sum, which represents <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{4}+T_{5}=5^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{4}+T_{5}=5^{2}}</annotation>

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</math></span><img src="./5488a65498b67c0cb3ee8d6b3c3926f4f5fb450c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.979ex; height:3.009ex;" alt="{\displaystyle T_{4}+T_{5}=5^{2}}" loading="lazy"></span> as <a href="Digit_sum" title="Digit sum">digit sums</a>:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{ccccccc}&4&3&2&1&\\+&1&2&3&4&5\\\hline &5&5&5&5&5\end{array}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ccccccc}&4&3&2&1&\\+&1&2&3&4&5\\\hline &5&5&5&5&5\end{array}}}</annotation>

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</math></span><img src="./0dd1218450c2debcde81b58b461ea71bad8e180a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:21.843ex; height:10.176ex;" alt="{\displaystyle {\begin{array}{ccccccc}&4&3&2&1&\\+&1&2&3&4&5\\\hline &5&5&5&5&5\end{array}}}" loading="lazy"></span>

</p><p>This fact can also be demonstrated graphically by positioning the triangles in opposite directions to create a square:

</p>

</div>

<p>The double of a triangular number, as in the visual proof from the above section <a href="#Formula">§ Formula</a>, is called a <a href="Pronic_number" title="Pronic number">pronic number</a>.

</p><p>There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36, 1225. Some of them can be generated by a simple recursive formula:

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<annotation encoding="application/x-tex">{\displaystyle S_{n+1}=4S_{n}\left(8S_{n}+1\right)}</annotation>

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</math></span></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}=1.}">

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<annotation encoding="application/x-tex">{\displaystyle S_{1}=1.}</annotation>

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</math></span><img src="./769837e6f47fab8d3a05dad584fd3503a31c3565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.387ex; height:2.509ex;" alt="{\displaystyle S_{1}=1.}" loading="lazy"></span>

</p><p><i>All</i> <a href="Square_triangular_number" title="Square triangular number">square triangular numbers</a> are found from the recursion

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<annotation encoding="application/x-tex">{\displaystyle S_{n}=34S_{n-1}-S_{n-2}+2}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle S_{0}=0}</annotation>

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</math></span><img src="./a60c168417a797edeccbb2990524bf29c0bd0acc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.74ex; height:2.509ex;" alt="{\displaystyle S_{0}=0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}=1.}">

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<annotation encoding="application/x-tex">{\displaystyle S_{1}=1.}</annotation>

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</p>

<p>Also, the <a href="Squared_triangular_number" title="Squared triangular number">square of the <span class="texhtml mvar" style="font-style:italic;">n</span>th triangular number</a> is the same as the sum of the cubes of the integers 1 to <span class="texhtml mvar" style="font-style:italic;">n</span>. This can also be expressed as

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.}</annotation>

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</p><p>The sum of the first <span class="texhtml mvar" style="font-style:italic;">n</span> triangular numbers is the <span class="texhtml mvar" style="font-style:italic;">n</span>th <a href="Tetrahedral_number" title="Tetrahedral number">tetrahedral number</a>:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}={\frac {n(n+1)(n+2)}{6}}.}</annotation>

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</p><p>More generally, the difference between the <span class="texhtml mvar" style="font-style:italic;">n</span>th <a href="Polygonal_number" title="Polygonal number"><span class="texhtml mvar" style="font-style:italic;">m</span>-gonal number</a> and the <span class="texhtml mvar" style="font-style:italic;">n</span>th <span class="texhtml">(<i>m</i> + 1)</span>-gonal number is the <span class="texhtml">(<i>n</i> − 1)</span>th triangular number. For example, the sixth <a href="Heptagonal_number" title="Heptagonal number">heptagonal number</a> (81) minus the sixth <a href="Hexagonal_number" title="Hexagonal number">hexagonal number</a> (66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any <a href="Centered_polygonal_number" title="Centered polygonal number">centered polygonal number</a>; the <span class="texhtml mvar" style="font-style:italic;">n</span>th centered <span class="texhtml mvar" style="font-style:italic;">k</span>-gonal number is obtained by the formula

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<annotation encoding="application/x-tex">{\displaystyle Ck_{n}=kT_{n-1}+1}</annotation>

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</p><p>where <span class="texhtml mvar" style="font-style:italic;">T</span> is a triangular number.

</p><p>The positive difference of two triangular numbers is a <a href="Trapezoidal_number" class="mw-redirect" title="Trapezoidal number">trapezoidal number</a>.

</p><p>The pattern found for triangular numbers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{2}+1}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{2}+1}{2}}}</annotation>

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</math></span><img src="./41f5af3d3a1eabc6b6bbebc1d1e35edcd27b7f3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.726ex; height:7.343ex;" alt="{\displaystyle \sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{2}+1}{2}}}" loading="lazy"></span> and for tetrahedral numbers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{3}+2}{3}},}">

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</math></span><img src="./62b722a8ced9667810be1fb5b967c5d81e5843e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.678ex; height:7.343ex;" alt="{\displaystyle \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{3}+2}{3}},}" loading="lazy"></span> which uses <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>, can be generalized. This leads to the formula:<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

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</p>

<div class="mw-heading mw-heading2"><h2 id="Other_properties">Other properties</h2></div>

<p>Triangular numbers correspond to the first-degree case of <a href="Faulhaber's_formula" title="Faulhaber's formula">Faulhaber's formula</a>.

</p>

<p>{{{annotations}}}

</p>

</div>

</div>

<div class="thumbcaption"><a href="Proof_without_words" title="Proof without words">Proof without words</a> that all hexagonal numbers are odd-sided triangular numbers</div>

</div></div>

<p>Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers.

</p><p>Every even <a href="Perfect_number" title="Perfect number">perfect number</a> is triangular (as well as hexagonal), given by the formula

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<annotation encoding="application/x-tex">{\displaystyle M_{p}2^{p-1}={\frac {M_{p}(M_{p}+1)}{2}}=T_{M_{p}}}</annotation>

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where <span class="texhtml mvar" style="font-style:italic;">M<sub>p</sub></span> is a <a href="Mersenne_prime" title="Mersenne prime">Mersenne prime</a>. No odd perfect numbers are known; hence, all known perfect numbers are triangular.

</p><p>For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.

</p><p>The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.

</p><p>In <a href="Base_10" class="mw-redirect" title="Base 10">base 10</a>, the <a href="Digital_root" title="Digital root">digital root</a> of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9:

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<p>The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9".

</p><p>The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.

</p><p>If <span class="texhtml mvar" style="font-style:italic;">x</span> is a triangular number, <span class="texhtml mvar" style="font-style:italic;">a</span> is an odd square, and <span class="texhtml"><i>b</i> = <span class="sfrac">⁠<span class="tion"><span class="num"><i>a</i> − 1</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span></span>, then <span class="texhtml"><i>ax</i> + <i>b</i></span> is also a triangular number. Note that <span class="texhtml mvar" style="font-style:italic;">b</span> will always be a triangular number, because <span class="texhtml">8<i>T<sub>n</sub></i> + 1 = (2<i>n</i> + 1)<sup>2</sup></span>, which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for <span class="texhtml mvar" style="font-style:italic;">b</span> given <span class="texhtml mvar" style="font-style:italic;">a</span> is an odd square is the inverse of this operation.

The first several pairs of this form (not counting <span class="texhtml">1<i>x</i> + 0</span>) are: <span class="texhtml">9<i>x</i> + 1</span>, <span class="texhtml">25<i>x</i> + 3</span>, <span class="texhtml">49<i>x</i> + 6</span>, <span class="texhtml">81<i>x</i> + 10</span>, <span class="texhtml">121<i>x</i> + 15</span>, <span class="texhtml">169<i>x</i> + 21</span>, ... etc. Given <span class="texhtml mvar" style="font-style:italic;">x</span> is equal to <span class="texhtml mvar" style="font-style:italic;">T<sub>n</sub></span>, these formulas yield <span class="texhtml"><i>T</i><sub>3<i>n</i> + 1</sub></span>, <span class="texhtml"><i>T</i><sub>5<i>n</i> + 2</sub></span>, <span class="texhtml"><i>T</i><sub>7<i>n</i> + 3</sub></span>, <span class="texhtml"><i>T</i><sub>9<i>n</i> + 4</sub></span>, and so on.

</p><p>The sum of the <a href="Multiplicative_inverse" title="Multiplicative inverse">reciprocals</a> of all the nonzero triangular numbers is

</mrow>

</mrow>

</munderover>

<mfrac>

<mfrac>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{1 \over {{n^{2}+n} \over 2}}=2\sum _{n=1}^{\infty }{1 \over {n^{2}+n}}=2.}</annotation>

</semantics>

</math></span></span>

</p><p>This can be shown by using the basic sum of a <a href="Telescoping_series" title="Telescoping series">telescoping series</a>:

</mrow>

</mrow>

</munderover>

<mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{1 \over {n(n+1)}}=1.}</annotation>

</semantics>

</math></span></span>

</p><p>In addition, the <i>n</i>th partial sum of this series can be written as:

<mfrac>

</mstyle>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2n \over {n+1}.}</annotation>

</semantics>

</math></span></span>

</p><p>Two other formulas regarding triangular numbers are

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{a+b}=T_{a}+T_{b}+ab}</annotation>

</semantics>

</math></span></span>

and

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{ab}=T_{a}T_{b}+T_{a-1}T_{b-1},}</annotation>

</semantics>

</math></span></span>

both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra. The first formula are relevant to <a href="Multiplication_algorithm#Quarter_square_multiplication" title="Multiplication algorithm">multiplication algorithm#Quarter square multiplication</a>.

</p><p>In 1796, Gauss discovered that every positive integer is representable as a sum of three triangular numbers, writing in his diary his famous words, "<a href="Eureka_(word)" title="Eureka (word)">ΕΥΡΗΚΑ!</a> <span class="nowrap">num = Δ + Δ + Δ</span>". The three triangular numbers are not necessarily distinct, or nonzero; for example 20 = 10 + 10 + 0. This is a special case of the <a href="Fermat_polygonal_number_theorem" title="Fermat polygonal number theorem">Fermat polygonal number theorem</a>.

</p><p>The largest triangular number of the form <span class="texhtml">2<sup><i>k</i></sup> − 1</span> is <a href="4000_(number)#4001_to_4099" title="4000 (number)">4095</a> (see <a href="Ramanujan%E2%80%93Nagell_equation" title="Ramanujan–Nagell equation">Ramanujan–Nagell equation</a>).

</p><p><a href="Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Franciszek Sierpiński</a> posed the question as to the existence of four distinct triangular numbers in <a href="Geometric_progression" title="Geometric progression">geometric progression</a>. It was conjectured by Polish mathematician Kazimierz Szymiczek to be impossible and was later proven by Fang and Chen in 2007.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p><p>Formulas involving expressing an integer as the sum of triangular numbers are connected to <a href="Theta_function" title="Theta function">theta functions</a>, in particular the <a href="Ramanujan_theta_function" title="Ramanujan theta function">Ramanujan theta function</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>

</p><p>The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a>:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{n}=3T_{n-1}=3{n \choose 2};\qquad L_{n}=L_{n-1}+3(n-1),~L_{1}=0.}</annotation>

</semantics>

</math></span></span>

</p><p>In the <a href="Limit_of_a_sequence" title="Limit of a sequence">limit</a>, the ratio between the two numbers, dots and line segments is

</mrow>

</munder>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {T_{n}}{L_{n}}}={\frac {1}{3}}.}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>

</p>

<p>The triangular number <span class="texhtml mvar" style="font-style:italic;">T<sub>n</sub></span> solves the <b>handshake problem</b> of counting the number of handshakes if each person in a room with <span class="texhtml"><i>n</i> + 1</span> people shakes hands once with each person. In other words, the solution to the handshake problem of <span class="texhtml mvar" style="font-style:italic;">n</span> people is <span class="texhtml"><i>T</i><sub><i>n</i>−1</sub></span>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p><p>Equivalently, a <a href="Fully_connected_network" class="mw-redirect" title="Fully connected network">fully connected network</a> of <span class="texhtml mvar" style="font-style:italic;">n</span> computing devices requires the presence of <span class="texhtml"><i>T</i><sub><i>n</i> − 1</sub></span> cables or other connections.

</p><p>A triangular number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{n}}</annotation>

</semantics>

</math></span><img src="./4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy"></span> is equivalent to the number of principal rotations in dimension <span class="mwe-math-element mwe-math-element-inline"><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+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

</semantics>

</math></span><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"></span>. For example, in five dimensions the number of principal rotations is 10 which is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{4}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{4}}</annotation>

</semantics>

</math></span><img src="./83a8e4649fd334cf176ee7badff8eb949ecd3670.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{4}}" loading="lazy"></span>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>

</p><p>In a tournament format that uses a round-robin <a href="Group_stage" class="mw-redirect" title="Group stage">group stage</a>, the number of matches that need to be played between <span class="texhtml mvar" style="font-style:italic;">n</span> teams is equal to the triangular number <span class="texhtml"><i>T</i><sub><i>n</i> − 1</sub></span>. For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.

</p>

<p>One way of calculating the <a href="Depreciation" title="Depreciation">depreciation</a> of an asset is the <a href="Depreciation#Sum-of-years-digits_method" title="Depreciation">sum-of-years' digits method</a>, which involves finding <span class="texhtml mvar" style="font-style:italic;">T<sub>n</sub></span>, where <span class="texhtml mvar" style="font-style:italic;">n</span> is the length in years of the asset's useful life. Each year, the item loses <span class="texhtml">(<i>b</i> − <i>s</i>) × <span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i> − <i>y</i></span><span class="sr-only">/</span><span class="den"><i>T<sub>n</sub></i></span></span>⁠</span></span>, where <span class="texhtml mvar" style="font-style:italic;">b</span> is the item's beginning value (in units of currency), <span class="texhtml mvar" style="font-style:italic;">s</span> is its final salvage value, <span class="texhtml mvar" style="font-style:italic;">n</span> is the total number of years the item is usable, and <span class="texhtml mvar" style="font-style:italic;">y</span> the current year in the depreciation schedule. Under this method, an item with a usable life of <span class="texhtml mvar" style="font-style:italic;">n</span> = 4 years would lose <span class="sfrac">⁠<span class="tion"><span class="num">4</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span> of its "losable" value in the first year, <span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span> in the second, <span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span> in the third, and <span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span> in the fourth, accumulating a total depreciation of <span class="sfrac">⁠<span class="tion"><span class="num">10</span><span class="sr-only">/</span><span class="den">10</span></span>⁠</span> (the whole) of the losable value.

</p><p><a href="Board_game" title="Board game">Board game</a> designers Geoffrey Engelstein and Isaac Shalev describe triangular numbers as having achieved "nearly the status of a mantra or koan among <a href="Game_design" title="Game design">game designers</a>", describing them as "deeply intuitive" and "featured in an enormous number of games, [proving] incredibly versatile at providing escalating rewards for larger sets without overly incentivizing specialization to the exclusion of all other strategies".<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p>

<caption>Relationship between the maximum number of pips on an end of a <a href="Domino" class="mw-redirect" title="Domino">domino</a> and the number of dominoes in its set<br>(values in bold are common)

</caption>

<th>Max. pips

</th>

<td>5

</td>

<th>6

</th>

<td>8

</td>

<th>9

</th>

<td>11

</td>

<th>12

</th>

<td>14

</td>

<th>15

</th>

<td>17

</td>

<th>18

</th>

<td>21

</td></tr>

<tr>

</th>

<td>6

</td>

<th>7

</th>

<td>9

</td>

<th>10

</th>

<td>12

</td>

<th>13

</th>

<td>15

</td>

<th>16

</th>

<td>18

</td>

<th>19

</th>

<td>22

</td></tr>

<tr>

</th>

<td>21

</td>

<th>28

</th>

<td>45

</td>

<th>55

</th>

<td>78

</td>

<th>91

</th>

<td>120

</td>

<th>136

</th>

<td>161

</td>

<th>190

</th>

<td>253

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Triangular_roots_and_tests_for_triangular_numbers">Triangular roots and tests for triangular numbers</h2></div>

<p>By analogy with the <a href="Square_root" title="Square root">square root</a> of <span class="texhtml mvar" style="font-style:italic;">x</span>, one can define the (positive) triangular root of <span class="texhtml mvar" style="font-style:italic;">x</span> as the number <span class="texhtml mvar" style="font-style:italic;">n</span> such that <span class="texhtml"><i>T<sub>n</sub></i> = <i>x</i></span>:<sup id="cite_ref-EulerRoots_18-0" class="reference"><a href="#cite_note-EulerRoots-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n={\frac {{\sqrt {8x+1}}-1}{2}}}</annotation>

</semantics>

</math></span></span>

</p><p>which follows immediately from the <a href="Quadratic_formula" title="Quadratic formula">quadratic formula</a>. So an integer <span class="texhtml mvar" style="font-style:italic;">x</span> is triangular <a href="If_and_only_if" title="If and only if">if and only if</a> <span class="texhtml">8<i>x</i> + 1</span> is a square. Equivalently, if the positive triangular root <span class="texhtml mvar" style="font-style:italic;">n</span> of <span class="texhtml mvar" style="font-style:italic;">x</span> is an integer, then <span class="texhtml mvar" style="font-style:italic;">x</span> is the <span class="texhtml mvar" style="font-style:italic;">n</span>th triangular number.<sup id="cite_ref-EulerRoots_18-1" class="reference"><a href="#cite_note-EulerRoots-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Alternative_name">Alternative name</h2></div>

<p>By analogy with the <a href="Factorial" title="Factorial">factorial</a> function, a product whose factors are the integers from 1 to <span class="texhtml mvar" style="font-style:italic;">n</span>, <a href="Donald_Knuth" title="Donald Knuth">Donald Knuth</a> proposed the name <b>Termial function</b>,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> with the notation <span class="texhtml mvar" style="font-style:italic;">n</span>? for the sum whose terms are the integers from 1 to <span class="texhtml mvar" style="font-style:italic;">n</span> (the <span class="texhtml mvar" style="font-style:italic;">n</span>th triangular number). Although some other sources use this name and notation,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> they are not in wide use.

</p>

<li><a href="Doubly_triangular_number" title="Doubly triangular number">Doubly triangular number</a>, a triangular number whose position in the sequence of triangular numbers is also a triangular number</li>

<li><a href="Tetractys" title="Tetractys">Tetractys</a>, an arrangement of ten points in a triangle, important in Pythagoreanism</li>

<li><a href="Factoriangular_number" title="Factoriangular number">Factoriangular number</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/algebra/triangular-numbers.html">"Triangular Number Sequence"</a>. <i>Math Is Fun</i>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFSpivak2008" class="citation book cs1"><a href="Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7JKVu_9InRUC&pg=PA21"><i>Calculus</i></a> (4th ed.). Houston, Texas: Publish or Perish. pp. <span class="nowrap">21–</span>22. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-914098-91-1</bdi>.</cite></span>

</li>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Triangular_numbers" class="extiw external" title="commons:Category:Triangular numbers">triangular numbers</a></span>.</div></div>

</div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Arithmetic_series">"Arithmetic series"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/do_you_know/numbers.shtml#square">Triangular numbers</a> at <a href="Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li>

<li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/do_you_know/triSquare.shtml">There exist triangular numbers that are also square</a> at <a href="Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li>

<li><span class="citation mathworld" id="Reference-Mathworld-Triangular_Number"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TriangularNumber.html">"Triangular Number"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>

<li><a rel="nofollow" class="external text" href="http://rhubbarb.wordpress.com/2009/04/23/hypertetrahedral-polytopic-roots/">Hypertetrahedral Polytopic Roots</a> by Rob Hubbard, including the generalisation to <i>triangular cube roots</i>, some higher dimensions, and some approximate formulas</li></ul>

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</style><div id="Figurate_numbers160" style="font-size:114%;margin:0 4em"><a href="Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_triangular_number" title="Centered triangular number">Centered triangular numbers</a></li>

<li><a href="Centered_square_number" title="Centered square number">Centered square numbers</a></li>

<li><a href="Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal numbers</a></li>

<li><a href="Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal numbers</a></li>

<li><a href="Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal numbers</a></li>

<li><a href="Centered_octagonal_number" title="Centered octagonal number">Centered octagonal numbers</a></li>

<li><a href="Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal numbers</a></li>

<li><a href="Centered_decagonal_number" title="Centered decagonal number">Centered decagonal numbers</a></li>

<li><a href="Star_number" title="Star number">Star numbers</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Square_number" title="Square number">Square numbers</a></li>

<li><a href="Pentagonal_number" title="Pentagonal number">Pentagonal numbers</a></li>

<li><a href="Hexagonal_number" title="Hexagonal number">Hexagonal numbers</a></li>

<li><a href="Heptagonal_number" title="Heptagonal number">Heptagonal numbers</a></li>

<li><a href="Octagonal_number" title="Octagonal number">Octagonal numbers</a></li>

<li><a href="Nonagonal_number" title="Nonagonal number">Nonagonal numbers</a></li>

<li><a href="Decagonal_number" title="Decagonal number">Decagonal numbers</a></li>

<li><a href="Dodecagonal_number" title="Dodecagonal number">Dodecagonal numbers</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral numbers</a></li>

<li><a href="Centered_cube_number" title="Centered cube number">Centered cube numbers</a></li>

<li><a href="Centered_octahedral_number" title="Centered octahedral number">Centered octahedral numbers</a></li>

<li><a href="Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral numbers</a></li>

<li><a href="Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral numbers</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cube_(algebra)" title="Cube (algebra)">Cube numbers</a></li>

<li><a href="Octahedral_number" title="Octahedral number">Octahedral numbers</a></li>

<li><a href="Dodecahedral_number" title="Dodecahedral number">Dodecahedral numbers</a></li>

<li><a href="Icosahedral_number" title="Icosahedral number">Icosahedral numbers</a></li>

<li><a href="Stella_octangula_number" title="Stella octangula number">Stella octangula numbers</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tetrahedral_number" title="Tetrahedral number">Tetrahedral numbers</a></li>

<li><a href="Square_pyramidal_number" title="Square pyramidal number">Square pyramidal numbers</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pentatope_number" title="Pentatope number">Pentatope numbers</a></li>

<li><a href="Squared_triangular_number" title="Squared triangular number">Squared triangular numbers</a></li>

<li><a href="Fourth_power" title="Fourth power">Tesseractic numbers</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Higher <a href="Dimension" title="Dimension">dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="non-centered12" scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Fifth_power_(algebra)" title="Fifth power (algebra)">5-hypercube numbers</a></li>

<li><a href="Sixth_power" title="Sixth power">6-hypercube numbers</a></li>

<li><a href="Seventh_power" title="Seventh power">7-hypercube numbers</a></li>

<li><a href="Eighth_power" title="Eighth power">8-hypercube numbers</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Sequences_and_series332" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Sequences_and_series332" style="font-size:114%;margin:0 4em"><a href="Sequence" title="Sequence">Sequences</a> and <a href="Series_(mathematics)" title="Series (mathematics)">series</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Integer_sequence" title="Integer sequence">Integer sequences</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Basic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a></li>

<li><a href="Geometric_progression" title="Geometric progression">Geometric progression</a></li>

<li><a href="Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">Harmonic progression</a></li>

<li><a href="Square_number" title="Square number">Square number</a></li>

<li><a href="Cube_(algebra)" title="Cube (algebra)">Cubic number</a></li>

<li><a href="Factorial" title="Factorial">Factorial</a></li>

<li><a href="Power_of_two" title="Power of two">Powers of two</a></li>

<li><a href="Power_of_three" title="Power of three">Powers of three</a></li>

<li><a href="Power_of_10" title="Power of 10">Powers of 10</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Advanced <span class="nobold">(<a href="List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">list</a>)</span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Complete_sequence" title="Complete sequence">Complete sequence</a></li>

<li><a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a></li>

<li><a href="Figurate_number" title="Figurate number">Figurate number</a></li>

<li><a href="Heptagonal_number" title="Heptagonal number">Heptagonal number</a></li>

<li><a href="Hexagonal_number" title="Hexagonal number">Hexagonal number</a></li>

<li><a href="Lucas_number" title="Lucas number">Lucas number</a></li>

<li><a href="Pell_number" title="Pell number">Pell number</a></li>

<li><a href="Pentagonal_number" title="Pentagonal number">Pentagonal number</a></li>

<li><a href="Polygonal_number" title="Polygonal number">Polygonal number</a></li>

<li>

<ul><li><a href="Triangular_array" title="Triangular array">array</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="Fibonacci_sequence" title="Fibonacci sequence"></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of sequences</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li>

<li><a href="Monotonic_function" title="Monotonic function">Monotonic function</a></li>

<li><a href="Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Series</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Convergent_series" title="Convergent series">Convergent</a></li>

<li><a href="Divergent_series" title="Divergent series">Divergent</a></li>

<li><a href="Telescoping_series" title="Telescoping series">Telescoping</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergence</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolute_convergence" title="Absolute convergence">Absolute</a></li>

<li><a href="Conditional_convergence" title="Conditional convergence">Conditional</a></li>

<li><a href="Uniform_convergence" title="Uniform convergence">Uniform</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicit series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergent</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Riemann_zeta_function" title="Riemann zeta function">1 + 1/2<sup><i>s</i></sup> + 1/3<sup><i>s</i></sup> + ... (Riemann zeta function)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Divergent</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Grandi's_series" title="Grandi's series">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li>

<li><a href="Infinite_arithmetic_series" class="mw-redirect" title="Infinite arithmetic series">Infinite arithmetic series</a></li>

<li><a href="1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_%E2%8B%AF" title="1 − 1 + 2 − 6 + 24 − 120 + ⋯">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)</a></li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)</a></li>

<li><a href="Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kinds of series</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Taylor_series" title="Taylor series">Taylor series</a></li>

<li><a href="Power_series" title="Power series">Power series</a></li>

<li><a href="Formal_power_series" title="Formal power series">Formal power series</a></li>

<li><a href="Laurent_series" title="Laurent series">Laurent series</a></li>

<li><a href="Puiseux_series" title="Puiseux series">Puiseux series</a></li>

<li><a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li>

<li><a href="Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li>

<li><a href="Fourier_series" title="Fourier series">Fourier series</a></li>

<li><a href="Generating_series" class="mw-redirect" title="Generating series">Generating series</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Hypergeometric_function" title="Hypergeometric function">Hypergeometric series</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">Generalized hypergeometric series</a></li>

<li><a href="Hypergeometric_function_of_a_matrix_argument" title="Hypergeometric function of a matrix argument">Hypergeometric function of a matrix argument</a></li>

<li><a href="Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li>

<li><a href="Modular_hypergeometric_series" class="mw-redirect" title="Modular hypergeometric series">Modular hypergeometric series</a></li>

<li><a href="Riemann's_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li>

<li><a href="Theta_hypergeometric_series" class="mw-redirect" title="Theta hypergeometric series">Theta hypergeometric series</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Classes_of_natural_numbers743" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Classes_of_natural_numbers743" style="font-size:114%;margin:0 4em">Classes of <a href="Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers743" style="font-size:114%;margin:0 4em"><a href="Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Achilles_number" title="Achilles number">Achilles</a></li>

<li><a href="Power_of_two" title="Power of two">Power of 2</a></li>

<li><a href="Power_of_three" title="Power of three">Power of 3</a></li>

<li><a href="Power_of_10" title="Power of 10">Power of 10</a></li>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Fourth_power" title="Fourth power">Fourth power</a></li>

<li><a href="Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li>

<li><a href="Sixth_power" title="Sixth power">Sixth power</a></li>

<li><a href="Seventh_power" title="Seventh power">Seventh power</a></li>

<li><a href="Eighth_power" title="Eighth power">Eighth power</a></li>

<li><a href="Perfect_power" title="Perfect power">Perfect power</a></li>

<li><a href="Powerful_number" title="Powerful number">Powerful</a></li>

<li><a href="Prime_power" title="Prime power">Prime power</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1743" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cullen_number" title="Cullen number">Cullen</a></li>

<li><a href="Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li>

<li><a href="Fermat_number" title="Fermat number">Fermat</a></li>

<li><a href="Mersenne_prime" title="Mersenne prime">Mersenne</a></li>

<li><a href="Proth_number" class="mw-redirect" title="Proth number">Proth</a></li>

<li><a href="Thabit_number" title="Thabit number">Thabit</a></li>

<li><a href="Woodall_number" title="Woodall number">Woodall</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers743" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hilbert_number" title="Hilbert number">Hilbert</a></li>

<li><a href="Idoneal_number" title="Idoneal number">Idoneal</a></li>

<li><a href="Leyland_number" title="Leyland number">Leyland</a></li>

<li><a href="Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li>

<li><a href="Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers743" style="font-size:114%;margin:0 4em"><a href="Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li>

<li><a href="Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li>

<li><a href="Leonardo_number" title="Leonardo number">Leonardo</a></li>

<li><a href="Lucas_number" title="Lucas number">Lucas</a></li>

<li><a href="Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li>

<li><a href="Padovan_sequence" title="Padovan sequence">Padovan</a></li>

<li><a href="Perrin_number" title="Perrin number">Perrin</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers743" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amenable_number" title="Amenable number">Amenable</a></li>

<li><a href="Congruent_number" title="Congruent number">Congruent</a></li>

<li><a href="Kn%C3%B6del_number" title="Knödel number">Knödel</a></li>

<li><a href="Riesel_number" title="Riesel number">Riesel</a></li>

<li><a href="Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums743" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li>

<li><a href="Polite_number" title="Polite number">Polite</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li>

<li><a href="Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers743" style="font-size:114%;margin:0 4em"><a href="Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li>

<li><a href="Centered_square_number" title="Centered square number">Centered square</a></li>

<li><a href="Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li>

<li><a href="Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li>

<li><a href="Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li>

<li><a href="Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li>

<li><a href="Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li>

<li><a href="Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Square_triangular_number" title="Square triangular number">Square triangular</a></li>

<li><a href="Pentagonal_number" title="Pentagonal number">Pentagonal</a></li>

<li><a href="Hexagonal_number" title="Hexagonal number">Hexagonal</a></li>

<li><a href="Heptagonal_number" title="Heptagonal number">Heptagonal</a></li>

<li><a href="Octagonal_number" title="Octagonal number">Octagonal</a></li>

<li><a href="Nonagonal_number" title="Nonagonal number">Nonagonal</a></li>

<li><a href="Decagonal_number" title="Decagonal number">Decagonal</a></li>

<li><a href="Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li>

<li><a href="Centered_cube_number" title="Centered cube number">Centered cube</a></li>

<li><a href="Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li>

<li><a href="Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li>

<li><a href="Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li>

<li><a href="Cube_(algebra)" title="Cube (algebra)">Cubic</a></li>

<li><a href="Octahedral_number" title="Octahedral number">Octahedral</a></li>

<li><a href="Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li>

<li><a href="Icosahedral_number" title="Icosahedral number">Icosahedral</a></li>

<li><a href="Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pentatope_number" title="Pentatope number">Pentatope</a></li>

<li><a href="Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li>

<li><a href="Fourth_power" title="Fourth power">Tesseractic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers743" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Catalan_number" title="Catalan number">Catalan</a></li>

<li><a href="Dedekind_number" title="Dedekind number">Dedekind</a></li>

<li><a href="Delannoy_number" title="Delannoy number">Delannoy</a></li>

<li><a href="Euler_number" class="mw-redirect" title="Euler number">Euler</a></li>

<li><a href="Eulerian_number" title="Eulerian number">Eulerian</a></li>

<li><a href="Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li>

<li><a href="Lazy_caterer's_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li>

<li><a href="Motzkin_number" title="Motzkin number">Motzkin</a></li>

<li><a href="Narayana_number" title="Narayana number">Narayana</a></li>

<li><a href="Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li>

<li><a href="Schr%C3%B6der_number" title="Schröder number">Schröder</a></li>

<li><a href="Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li>

<li><a href="Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li>

<li><a href="Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li>

<li><a href="Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li>

<li><a href="Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes743" style="font-size:114%;margin:0 4em"><a href="Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li>

<li><a href="Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li>

<li><a href="Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes743" style="font-size:114%;margin:0 4em"><a href="Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carmichael_number" title="Carmichael number">Carmichael number</a></li>

<li><a href="Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li>

<li><a href="Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li>

<li><a href="Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li>

<li><a href="Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li>

<li><a href="Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li>

<li><a href="Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li>

<li><a href="Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li>

<li><a href="Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li>

<li><a href="Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li>

<li><a href="Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics743" style="font-size:114%;margin:0 4em"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abundant_number" title="Abundant number">Abundant</a></li>

<li><a href="Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li>

<li><a href="Arithmetic_number" title="Arithmetic number">Arithmetic</a></li>

<li><a href="Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li>

<li><a href="Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li>

<li><a href="Deficient_number" title="Deficient number">Deficient</a></li>

<li><a href="Descartes_number" title="Descartes number">Descartes</a></li>

<li><a href="Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li>

<li><a href="Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li>

<li><a href="Highly_composite_number" title="Highly composite number">Highly composite</a></li>

<li><a href="Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li>

<li><a href="Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li>

<li><a href="Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li>

<li><a href="Refactorable_number" title="Refactorable number">Refactorable</a></li>

<li><a href="Semiperfect_number" title="Semiperfect number">Semiperfect</a></li>

<li><a href="Sublime_number" title="Sublime number">Sublime</a></li>

<li><a href="Superabundant_number" title="Superabundant number">Superabundant</a></li>

<li><a href="Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li>

<li><a href="Superperfect_number" title="Superperfect number">Superperfect</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost_prime" title="Almost prime">Almost prime</a></li>

<li><a href="Semiprime" title="Semiprime">Semiprime</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li>

<li><a href="Highly_totient_number" title="Highly totient number">Highly totient</a></li>

<li><a href="Noncototient" title="Noncototient">Noncototient</a></li>

<li><a href="Nontotient" title="Nontotient">Nontotient</a></li>

<li><a href="Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li>

<li><a href="Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amicable_numbers" title="Amicable numbers">Amicable</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li>

<li><a href="Untouchable_number" title="Untouchable number">Untouchable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euclid_number" title="Euclid number">Euclid</a></li>

<li><a href="Fortunate_number" title="Fortunate number">Fortunate</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers743" style="font-size:114%;margin:0 4em">Other <a href="Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li>

<li><a href="Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li>

<li><a href="Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li>

<li><a href="Friendly_number" title="Friendly number">Friendly</a></li>

<li><a href="Giuga_number" title="Giuga number">Giuga</a></li>

<li><a href="Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li>

<li><a href="Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li>

<li><a href="Pronic_number" title="Pronic number">Pronic</a></li>

<li><a href="Regular_number" title="Regular number">Regular</a></li>

<li><a href="Rough_number" title="Rough number">Rough</a></li>

<li><a href="Smooth_number" title="Smooth number">Smooth</a></li>

<li><a href="Sphenic_number" title="Sphenic number">Sphenic</a></li>

<li><a href="St%C3%B8rmer_number" title="Størmer number">Størmer</a></li>

<li><a href="Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers743" style="font-size:114%;margin:0 4em"><a href="Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br>and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Persistence_of_a_number" title="Persistence of a number">Persistence</a>

<ul><li><a href="Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li>

<li><a href="Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Digit_sum" title="Digit sum">Digit sum</a></li>

<li><a href="Digital_root" title="Digital root">Digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Meertens_number" title="Meertens number">Meertens</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dudeney_number" title="Dudeney number">Dudeney</a></li>

<li><a href="Factorion" title="Factorion">Factorion</a></li>

<li><a href="Kaprekar_number" title="Kaprekar number">Kaprekar</a></li>

<li><a href="Kaprekar's_routine" title="Kaprekar's routine">Kaprekar's constant</a></li>

<li><a href="Keith_number" title="Keith number">Keith</a></li>

<li><a href="Lychrel_number" title="Lychrel number">Lychrel</a></li>

<li><a href="Narcissistic_number" title="Narcissistic number">Narcissistic</a></li>

<li><a href="Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li>

<li><a href="Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a>

<ul><li><a href="Happy_number" title="Happy number">Happy</a></li></ul></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Automorphic_number" title="Automorphic number">Automorphic</a>

<ul><li><a href="Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Palindromic_number" title="Palindromic number">Palindromic</a></li>

<li><a href="Pandigital_number" title="Pandigital number">Pandigital</a></li>

<li><a href="Repdigit" title="Repdigit">Repdigit</a></li>

<li><a href="Repunit" title="Repunit">Repunit</a></li>

<li><a href="Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li>

<li><a href="Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li>

<li><a href="Undulating_number" title="Undulating number">Undulating</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cyclic_number" title="Cyclic number">Cyclic</a></li>

<li><a href="Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li>

<li><a href="Parasitic_number" title="Parasitic number">Parasitic</a></li>

<li><a href="Primeval_number" title="Primeval number">Primeval</a></li>

<li><a href="Transposable_integer" title="Transposable integer">Transposable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Equidigital_number" title="Equidigital number">Equidigital</a></li>

<li><a href="Extravagant_number" title="Extravagant number">Extravagant</a></li>

<li><a href="Frugal_number" title="Frugal number">Frugal</a></li>

<li><a href="Harshad_number" title="Harshad number">Harshad</a></li>

<li><a href="Polydivisible_number" title="Polydivisible number">Polydivisible</a></li>

<li><a href="Smith_number" title="Smith number">Smith</a></li>

<li><a href="Vampire_number" title="Vampire number">Vampire</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Friedman_number" title="Friedman number">Friedman</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers743" style="font-size:114%;margin:0 4em"><a href="Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Odious_number" title="Odious number">Odious</a></li>

<li><a href="Pernicious_number" title="Pernicious number">Pernicious</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve743" style="font-size:114%;margin:0 4em">Generated via a <a href="Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lucky_number" title="Lucky number">Lucky</a></li>

<li><a href="Generation_of_primes" title="Generation of primes">Prime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related743" style="font-size:114%;margin:0 4em"><a href="Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pancake_sorting" title="Pancake sorting">Pancake number</a></li>

<li><a href="Sorting_number" title="Sorting number">Sorting number</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related743" style="font-size:114%;margin:0 4em"><a href="Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aronson's_sequence" title="Aronson's sequence">Aronson's sequence</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related743" style="font-size:114%;margin:0 4em"><a href="Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div>

<ul><li><span class="noviewer" typeof="mw:File"></span> <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-07-27" href="https://en.wikipedia.org/wiki/?title=Triangular_number&oldid=1302809483">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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