已知 ΔABC 周长为 p，边长 BC=a，CA=b，AB=c 都是整数，求 (a,b,c) 的不同的取法总数

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luyuanhong

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王守恩

luyuanhong 发表于 2021-8-12 18:43

1，已知三角形每条边的边长皆为整数，且周界为2015。求三角形边长的可能组合的数目.
{0, 0, 1, 0, 3, 1, 6, 3, 10, 6, 15, 10, 21, 15, 28, 21, 36, 28, 45, 36, 55, 45, 66, 55, 78, 66,91,78,
105, 91, 120, 105, 136, 120,153, 136, 171,153, 190, 171, 210, 190, 231, 210, 253, 231, 276,
253, 300, 276, 325, 300, 351, 325, 378, 351, 406, 378, 435, 406, 465, 435, 496, 465, 528, ....}

\(a(n)=\frac{(2n-3\cos(n\pi)-1)(2n-3\cos(n\pi)-5)}{32}\)
特别地，\(a(2015)=507528\)

2，已知三角形每条边的边长皆为整数，且周界为2015。求三角形边长的可能组合的数目(各种排列只计算1种).
{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, ......}

\(a(n)=[\frac{(2n-3\cos(n\pi)+3)^2}{192}]\ \ \ [\ \ ]\)表示四舍五入

或：\(a(n)=\frac{6n^2+18n-9(2n+3)\cos(n\pi)-36\sin(n\pi/2)-36\cos(n\pi/2)+64\cos(2n\pi/3)-1}{288}\)

特别地，\(a(2015)=84840\)

3，可以归纳到一个公式上来。
已知 a 边形每条边的边长皆为整数，且周界为 n。求 a 边形边长的可能组合的数目(各种排列只计算1种).

CoefficientList[Series[\(\frac{x^a}{\prod_{k = 1}^a(1 - x^k)} -\frac{ 1}{\prod_{k = 1}^{a-1}(1 - x^{2k})}*\frac{x^{2 a - 2}}{1 - x}\),{x,0,n}],x]

3边形：{0, 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,}
4边形：{0, 0, 0, 0, 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,}
5边形：{0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 16, 23, 25, 35, 39, 52, 57, 74, 81, 103, 111, 139, 150, 184,}
6边形：{0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 28, 37, 46, 59, 71, 91, 107,134, 157,193, 222, 271,}
7边形：{0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 34, 42, 58, 70, 93, 112, 145,171,218, 256, 320,}
8边形：{0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 104, 134, 167, 211, 258, 322,}
9边形：{0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11,14, 21, 28, 39, 50, 69, 87, 116, 145,189,233,299, 363,}
........
n=0,1,2,3,4,5,6,7,8,.....