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发表于 2021-11-1 14:57
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本帖最后由 王守恩 于 2021-11-1 15:06 编辑
谢谢 Nicolas2050!
挑战一下:做题目的最高境界,一次到位,还敢有吗?!
2,将正整数 n 拆分成 2 个不全相等的正整数相加,有a(n)种不同的拆分法。
a(n)=CoefficientList\(\bigg[\)Series\(\bigg[\)\(\displaystyle\prod_{i=1}^2\frac{1}{1-x^i},(x,0,n)\bigg],x\bigg]+\bigg\lceil\frac{n_{2}}{2}\bigg\rceil-1\) n=2,3,4,5,......
{0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16,
3,将正整数 n 拆分成 3 个不全相等的正整数相加,有a(n)种不同的拆分法。
a(n)=CoefficientList\(\bigg[\)Series\(\bigg[\)\(\displaystyle\prod_{i=1}^3\frac{1}{1-x^i},(x,0,n)\bigg],x\bigg]+\bigg\lceil\frac{n_{3}}{3}\bigg\rceil-1\) n=3,4,5,6,......
{0, 1, 2, 2, 4, 5, 6, 8, 10, 11, 14, 16, 18, 21, 24, 26, 30, 33, 36, 40, 44, 47, 52, 56, 60, 65, 70, 74, 80,
4,将正整数 n 拆分成 4 个不全相等的正整数相加,有a(n)种不同的拆分法。
a(n)=CoefficientList\(\bigg[\)Series\(\bigg[\)\(\displaystyle\prod_{i=1}^4\frac{1}{1-x^i},(x,0,n)\bigg],x\bigg]+\bigg\lceil\frac{n_{4}}{4}\bigg\rceil-1\) n=4,5,6,7,......
{0, 1, 2, 3, 4, 6, 9, 11, 14, 18, 23, 27, 33, 39, 47, 54, 63, 72, 84, 94, 107, 120, 136, 150, 168, 185, 206,
5,将正整数 n 拆分成 5 个不全相等的正整数相加,有a(n)种不同的拆分法。
a(n)=CoefficientList\(\bigg[\)Series\(\bigg[\)\(\displaystyle\prod_{i=1}^5\frac{1}{1-x^i},(x,0,n)\bigg],x\bigg]+\bigg\lceil\frac{n_{5}}{5}\bigg\rceil-1\) n=5,6,7,8,......
{0, 1, 2, 3, 5, 6, 10, 13, 18, 23, 29, 37, 47, 57, 70, 83, 101, 119, 141, 164, 191, 221, 255, 291, 333, 376,
6,将正整数 n 拆分成 6 个不全相等的正整数相加,有a(n)种不同的拆分法。
a(n)=CoefficientList\(\bigg[\)Series\(\bigg[\)\(\displaystyle\prod_{i=1}^6\frac{1}{1-x^i},(x,0,n)\bigg],x\bigg]+\bigg\lceil\frac{n_{6}}{6}\bigg\rceil-1\) n=6,7,8,9,......
{0, 1, 2, 3, 5, 7, 10, 14, 20, 26, 35, 44, 57, 71, 90, 110, 136, 163, 198, 235, 282, 331, 391, 454, 531, 612,
7,将正整数 n 拆分成 7 个不全相等的正整数相加,有a(n)种不同的拆分法。
a(n)=CoefficientList\(\bigg[\)Series\(\bigg[\)\(\displaystyle\prod_{i=1}^7\frac{1}{1-x^i},(x,0,n)\bigg],x\bigg]+\bigg\lceil\frac{n_{7}}{7}\bigg\rceil-1\) n=7,8,9,10,......
{0, 1, 2, 3, 5, 7, 11, 14, 21, 28, 38, 49, 65, 82, 104, 131, 164, 201, 248, 300, 364, 435, 522, 618, 733, 860,
8,将正整数 n 拆分成 8 个不全相等的正整数相加,有a(n)种不同的拆分法。
a(n)=CoefficientList\(\bigg[\)Series\(\bigg[\)\(\displaystyle\prod_{i=1}^8\frac{1}{1-x^i},(x,0,n)\bigg],x\bigg]+\bigg\lceil\frac{n_{8}}{8}\bigg\rceil-1\) n=8,9,10,11,......
{0, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 52, 70, 89, 116, 146, 185, 230, 288, 352, 434, 525, 638, 764, 918, 1090,
9,将正整数 n 拆分成 9 个不全相等的正整数相加,有a(n)种不同的拆分法。
a(n)=CoefficientList\(\bigg[\)Series\(\bigg[\)\(\displaystyle\prod_{i=1}^9\frac{1}{1-x^i},(x,0,n)\bigg],x\bigg]+\bigg\lceil\frac{n_{9}}{9}\bigg\rceil-1\) n=9,10,11,12,......
{0, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 73, 94, 123, 157, 201, 252, 317, 393, 488, 598, 732, 887,1076,1291,
就这么些数字串,可是在《整数序列在线百科全书(OEIS)》找不到的。 |
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