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楼主: 王守恩

有多少个周长为 n 、三边边长都是整数的三角形?

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 楼主| 发表于 2023-5-26 16:16 | 显示全部楼层
周长为n, 有a(n)个3边为整数的3边形。
Table[Length@Select[IntegerPartitions[n, {3}], Max[#]2<n &], {n, 3, 100}]
{1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4,7, 5, 8, 7,10, 8, 12, 10, 14, 12, 16,14, 19, 16,21, 19, 24, 21, 27,  

周长为n, 有a(n)个4边为整数的4边形。
Table[Length@Select[IntegerPartitions[n, {4}], Max[#]2<n &], {n, 4, 100}]
{1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 12, 16, 18, 23, 24, 31, 33, 41, 43, 53, 55, 67, 69, 83, 86, 102, 104,

周长为n, 有a(n)个5边为整数的5边形。
Table[Length@Select[IntegerPartitions[n, {5}], Max[#]2<n &], {n, 5, 100}]
{1, 1, 2, 2, 4, 5, 8, 9, 14, 16, 23, 25, 35, 39, 52, 57, 74, 81, 103, 111, 139, 150, 184, 197, 239,

周长为n, 有a(n)个6边为整数的6边形。
Table[Length@Select[IntegerPartitions[n, {6}], Max[#]2<n &], {n, 6, 100}]
{1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 28, 37, 46,  59, 71,  91, 107, 134, 157, 193, 222, 271, 308, 371,

周长为n, 有a(n)个7边为整数的7边形。
Table[Length@Select[IntegerPartitions[n, {7}], Max[#]2<n &], {n, 7, 100}]
{1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 34, 42, 58, 70, 93, 112, 145, 171, 218, 256, 320, 372, 458, 528,

周长为n, 有a(n)个8边为整数的8边形。
Table[Length@Select[IntegerPartitions[n, {8}], Max[#]2<n &], {n, 8, 100}]
{1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 104, 134, 167, 211, 258, 322, 389, 480, 572, 698,   

周长为n, 有a(n)个9边为整数的9边形。
Table[Length@Select[IntegerPartitions[n, {9}], Max[#]2<n &], {n, 9, 100}]
{1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 50, 69, 87, 116, 145, 189, 233, 299, 363, 458, 553, 687, 820,

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謝謝老師  发表于 2024-10-27 10:19

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参与人数 1威望 +20 收起 理由
wlc1 + 20

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 楼主| 发表于 2024-7-1 10:44 | 显示全部楼层
每个{ }都是三角形。
a(1)=00,
a(2)=00,
a(3)=00,
a(4)=01, {2,3,4},
a(5)=02, {2,4,5},{3,4,5},
a(6)=04, {2,5,6},{3,4,6},{3,5,6},{4,5,6},
a(7)=06, {2,6,7},{3,5,7},{3,6,7},{4,5,7},{4,6,7},{5,6,7},
a(8)=09, {2,7,8},{3,6,8},{3,7,8},{4,5,8},{4,6,8},{4,7,8},{5,6,8},{5,7,8},{6,7,8},
a(9)=12, {2,8,9},{3,7,9},{3,8,9},{4,6,9},{4,7,9},{4,8,9},{5,6,9},{5,7,9},{5,8,9},{6,7,9},{6,8,9},{7,8,9},
a(0)=16, {2,9,0},{3,8,0},{3,9,0},{4,7,0},{4,8,0},{4,9,0},{5,6,0},{5,7,0},{5,8,0},{5,9,0},{6,7,0},{6,8,0},{6,9,0},{7,8,0},{7,9,0},{8,9,0},
a(1)=20, {2,0,1},{3,9,1},{3,0,1},{4,8,1},{4,9,1},{4,0,1},{5,7,1},{5,8,1},{5,9,1},{5,0,1},{6,7,1},{6,8,1},{6,9,1},{6,0,1},{7,8,1},{7,9,1},{7,0,1},{8,9,1},{8,0,1},{9,0,1},
a(1)=25, {2,1,2},{3,0,2},{3,1,2},{4,9,2},{4,0,2},{4,1,2},{5,8,2},{5,9,2},{5,0,2},{5,1,2},{6,7,2},{6,8,2},{6,9,2},{6,0,2},{6,1,2},{7,8,2},{7,9,2},{7,0,2},{7,1,2},{8,9,2},{8,0,2},{8,1,2},{9,0,2},{9,1,2},{0,1,2},
......

{0, 0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420, 441, 462, 484, 506, 529, 552, 576, 600, 625}
  1. Table[(n^2 - Mod[n, 2])/4, {n, 0, 50}]
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1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715, 825, 946, 1078, 1222, 1378, 1547, 1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843, 8372,
  1. Table[(n (n + 2) (2 n - 1) - 3 Mod[n, 2])/24, {n, 1, 49}]
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点评

謝謝老師  发表于 2024-10-27 10:19

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参与人数 1威望 +20 收起 理由
wlc1 + 20

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 楼主| 发表于 2025-3-15 06:22 | 显示全部楼层
有多少个周长n(n=1, 2, 3, 4, 5, 6, 7, 8, 9, ...),   整数边长(可以不等), 等角(必须相等)的六边形。
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\frac{\square}{\square}\sqrt{\square}\square_{\baguet}^{\baguet}\overarc{\square}\ \dot{\baguet}\left(\square\right)\binom{\square}{\square}\begin{cases}\square\\\square\end{cases}\ \begin{bmatrix}\square&\square\\\square&\square\end{bmatrix}\to\Rightarrow\mapsto\alpha\ \theta\ \pi\times\div\pm\because\angle\ \infty
\frac{\square}{\square}\sqrt{\square}\sqrt[\baguet]{\square}\square_{\baguet}\square^{\baguet}\square_{\baguet}^{\baguet}\sum_{\baguet}^{\baguet}\prod_{\baguet}^{\baguet}\coprod_{\baguet}^{\baguet}\int_{\baguet}^{\baguet}\lim_{\baguet}\lim_{\baguet}^{\baguet}\bigcup_{\baguet}^{\baguet}\bigcap_{\baguet}^{\baguet}\bigwedge_{\baguet}^{\baguet}\bigvee_{\baguet}^{\baguet}
\underline{\square}\overline{\square}\overrightarrow{\square}\overleftarrow{\square}\overleftrightarrow{\square}\underrightarrow{\square}\underleftarrow{\square}\underleftrightarrow{\square}\dot{\baguet}\hat{\baguet}\vec{\baguet}\tilde{\baguet}
\left(\square\right)\left[\square\right]\left\{\square\right\}\left|\square\right|\left\langle\square\right\rangle\left\lVert\square\right\rVert\left\lfloor\square\right\rfloor\left\lceil\square\right\rceil\binom{\square}{\square}\boxed{\square}
\begin{cases}\square\\\square\end{cases}\begin{matrix}\square&\square\\\square&\square\end{matrix}\begin{pmatrix}\square&\square\\\square&\square\end{pmatrix}\begin{bmatrix}\square&\square\\\square&\square\end{bmatrix}\begin{Bmatrix}\square&\square\\\square&\square\end{Bmatrix}\begin{vmatrix}\square&\square\\\square&\square\end{vmatrix}\begin{Vmatrix}\square&\square\\\square&\square\end{Vmatrix}\begin{array}{l|l}\square&\square\\\hline\square&\square\end{array}
\to\gets\leftrightarrow\nearrow\searrow\downarrow\uparrow\updownarrow\swarrow\nwarrow\Leftarrow\Rightarrow\Leftrightarrow\rightharpoonup\rightharpoondown\impliedby\implies\Longleftrightarrow\leftharpoonup\leftharpoondown\longleftarrow\longrightarrow\longleftrightarrow\Uparrow\Downarrow\Updownarrow\hookleftarrow\hookrightarrow\mapsto
\alpha\beta\gamma\Gamma\delta\Delta\epsilon\varepsilon\zeta\eta\theta\Theta\iota\kappa\varkappa\lambda\Lambda\mu\nu\xi\Xi\pi\Pi\varpi\rho\varrho\sigma\Sigma\tau\upsilon\Upsilon\phi\Phi\varphi\chi\psi\Psi\omega\Omega\digamma\vartheta\varsigma\mathbb{C}\mathbb{H}\mathbb{N}\mathbb{P}\mathbb{Q}\mathbb{R}\mathbb{Z}\Re\Im\aleph\partial\nabla
\times\cdot\ast\div\pm\mp\circ\backslash\oplus\ominus\otimes\odot\bullet\varnothing\neq\equiv\not\equiv\sim\approx\simeq\cong\geq\leq\ll\gg\succ\prec\in\ni\cup\cap\subset\supset\not\subset\not\supset\notin\not\ni\subseteq\supseteq\nsubseteq\nsupseteq\sqsubset\sqsupset\sqsubseteq\sqsupseteq\sqcap\sqcup\wedge\vee\neg\forall\exists\nexists\uplus\bigsqcup\bigodot\bigotimes\bigoplus\biguplus\bigcap\bigcup\bigvee\bigwedge
\because\therefore\angle\parallel\perp\top\nparallel\measuredangle\sphericalangle\diamond\diamondsuit\doteq\propto\infty\bowtie\square\smile\frown\bigtriangledown\triangle\triangleleft\triangleright\bigcirc \wr\amalg\models\preceq\mid\nmid\vdash\dashv\nless\ngtr\ldots\cdots\vdots\ddots\surd\ell\flat\sharp\natural\wp\clubsuit\heartsuit\spadesuit\oint\lfloor\rfloor\lceil\rceil\lbrace\rbrace\lbrack\rbrack\vert\hbar\aleph\dagger\ddagger

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