本帖最后由 王守恩 于 2021-8-13 19:26 编辑
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,}
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n=0,1,2,3,4,5,6,7,8,..... |