数学中国

标题: 有多少个周长为 n 、三边边长都是整数的三角形？ [打印本页]

作者: 王守恩 时间: 2022-10-27 20:11
标题: 有多少个周长为 n 、三边边长都是整数的三角形？
周长为n,有多少个三边为整数的三角形？

{0, 0, 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, 24, 30, 27, 33, 30, 37, 33, 40, 37, 44, 40, 48, 44, 52, 48, 56, 52, 61, 56,
65, 61, 70, 65, 75, 70, 80, 75, 85, 80, 91, 85, 96, 91, 102, 96, 108, 102, 114, 108, 120,
114, 127, 120, 133, 127, 140, 133, 147, 140, 154, 147, 161, 154, 169, 161, 176, 169,...

a(n)=[\(\frac{(n + 3 Mod(n, 2))^2}{48}\)] [ ]表示四舍五入

a(n)=SeriesCoefficient[\(\frac{x^3}{(1 - x^2) (1 - x^3) (1 - x^4)}\) {x, 0, n}]

作者: 王守恩 时间: 2022-10-28 08:33
你找到规律了吗？

a(1)=0
a(2)=0
a(3)=1:1,1,1,
a(4)=0
a(5)=1:1,2,2,
a(6)=1:2,2,2,
a(7)=2:1,3,3, 2,2,3,
a(8)=1:2,3,3,
a(9)=3:1,4,4, 2,3,4, 3,3,3,
a(10)=2:2,4,4, 3,3,4,
a(11)=4:1,5,5, 2,4,5, 3,3,5, 3,4,4,
a(12)=3:2,5,5, 3,4,5, 4,4,4,
a(13)=5:1,6,6, 2,5,6, 3,4,6, 3,5,5, 4,4,5,
a(14)=4:2,6,6, 3,5,6, 4,4,6, 4,5,5,
a(15)=7:1,7,7, 2,6,7, 3,5,7, 3,6,6, 4,4,7, 4,5,6, 5,5,5,
a(16)=5:2,7,7, 3,6,7, 4,5,7, 4,6,6, 5,5,6,
a(17)=8:1,8,8, 2,7,8, 3,6,8, 3,7,7, 4,4,7, 4,5,8, 4,6,7, 5,6,6,
a(18)=7:2,8,8, 3,7,8, 4,6,8, 4,7,7, 5,5,8, 5,6,7, 6,6,6,
a(19)=10:1,9,9, 2,8,9, 3,7,9, 3,8,8, 4,6,9, 4,7,8, 5,5,9, 5,6,8, 5,7,7, 6,6,7,
a(20)=08:2,9,9, 3,8,9, 4,7,9, 4,8,8, 5,6,9, 5,7,8, 6,6,8, 6,7,7,
a(21)=12:1,0,0, 2,9,0, 3,8,0, 3,9,9, 4,7,0, 4,8,9, 5,6,0, 5,7,9, 5,8,8, 6,6,9, 6,7,8, 7,7,7,
a(22)=10:2,0,0, 3,9,0, 4,8,0, 4,9,9, 5,7,0, 5,8,9, 6,6,0, 6,7,9, 6,8,8, 7,8,8,
a(23)=14:1,1,1, 2,0,1, 3,9,1, 3,0,0, 4,8,1, 4,9,0, 5,7,1, 5,8,0, 5,9,9, 6,6,1, 6,7,0, 6,8,9, 7,7,9, 7,8,8,
a(24)=12
a(25)=16
a(26)=14
a(27)=19
a(28)=16
a(29)=21
a(30)=19
a(31)=24
a(32)=21
a(33)=27
a(34)=24
a(35)=30
a(36)=27
a(37)=33
a(38)=30
a(39)=37
a(40)=33
a(41)=
a(42)=
a(43)=
a(44)=
a(45)=
a(46)=
a(47)=
a(48)=
a(49)=
a(50)=

作者: luyuanhong 时间: 2022-10-28 12:15
下面是我过去在《数学中国》发表过的一个帖子，可供参考：

[attach]118866[/attach]

[attach]118867[/attach]

作者: luyuanhong 时间: 2022-10-28 12:20
下面是我过去在《数学中国》发表过的一个帖子，可供参考：

[attach]118868[/attach]

[attach]118869[/attach]

作者: luyuanhong 时间: 2022-10-28 17:37
下面是我过去在《数学中国》发表过的一个帖子：

[attach]118874[/attach]
[attach]118875[/attach]

[attach]118877[/attach]

作者: luyuanhong 时间: 2022-10-28 17:42
[attach]118878[/attach]

[attach]118879[/attach]

作者: 王守恩 时间: 2022-10-30 13:46

王守恩发表于 2022-10-28 08:33
你找到规律了吗？

a(1)=0

规律很简单。

第1个数：从1,2,3,4...挨次找；
第2个数：不能比第1个数小，挨次找；
第3个数：不能比第2个数小，挨次找，注意比前2数之和小。

作者: 王守恩 时间: 2023-1-6 14:05
周长为n,有多少个三边为整数的三角形？

{0, 0, 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, 24, 30, 27, 33, 30, 37, 33, 40, 37, 44, 40, 48, 44, 52, 48, 56, 52, 61, 56,
65, 61, 70, 65, 75, 70, 80, 75, 85, 80, 91, 85, 96, 91, 102, 96, 108, 102, 114, 108, 120,
114, 127, 120, 133, 127, 140, 133, 147, 140, 154, 147, 161, 154, 169, 161, 176, 169,...

Table[Length@Select[IntegerPartitions[n, {3}], Max[#]*180 < 90 n &], {n, 1, 100}]

(* Frank M Jackson, Nov 04 2022 *)

这是2022年11月04日出来的通项公式！

2022年10月30日的时候还没有这个通项公式！

你不感觉应该努力追赶吗？你不感觉可以努力追赶吗？

作者: xfhaoym 时间: 2023-1-9 10:31
一个三角形的法则：任意两个边之和大于第三边，而任意两个边之差小于第三边？忘了。

作者: 王守恩 时间: 2023-1-9 10:58
不等边(边长为整数)三角形，有 a(n) 个最大边长为 n 的三角形。

OEIS有这串数，可惜没有这种说法。

作者: 王守恩 时间: 2023-5-26 16:16
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,

作者: 王守恩 时间: 2024-7-1 10:44
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}

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,

Table[(n (n + 2) (2 n - 1) - 3 Mod[n, 2])/24, {n, 1, 49}]

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作者: 王守恩 时间: 2025-3-15 06:22
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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