有多少个周长为 n 、三边边长都是整数的三角形？

王守恩

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,

王守恩

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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王守恩

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。