数字串的通项公式

王守恩

northwolves 发表于 2025-3-18 22:14
{0,0,1,2,3,4,6,8,10,12,14,16,19,22,25,28,31,34,37,40,44,48,52,56,60,64,68,72,76,80,85,90,95,100, ...

会有解吗？——a,b,c,d,x,r都是正整数——烧脑的几何难题(4): 四边形内切等圆问题该如何解决？——知乎

Table[NSolve[{Sqrt[((a + b)^2 - x^2) (x^2 - (a - b)^2)]/(2 (a + b + x)) == Sqrt[((c + d)^2 - x^2) (x^2 - (c - d)^2)]/(2 (c + d + x)) == r, r > 0, x > 0}, {x, r}, Integers], {a, 5}, {b, a, 9}, {c, a, 12}, {d, c, b - 1}]

王守恩

备忘。

{0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11},
Table[k, {n, 0, 14}, {k, 0, n}] // Flatten

{1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12},
Table[k, {n, 14}, {k, n}] // Flatten

{1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12,12},
Table[n, {n, 14}, {k, n}] // Flatten

{0, 1, 1, 3, 3, 3, 6, 6, 6, 6, 10, 10, 10, 10, 10, 15, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21, 21, 21, 28, 28, 28, 28, 28, 28, 28, 28, 36, 36, 36, 36, 36, 36, 36, 36, 36, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 55, 55, 55, 55},
Table[n (n - 1)/2, {n, 15}, {n}] // Flatten

{1, 3, 3, 6, 6, 6, 10, 10, 10, 10, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21, 21, 28, 28, 28, 28, 28, 28, 28, 36, 36, 36, 36, 36, 36, 36, 36, 45, 45, 45, 45, 45, 45, 45, 45, 45, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 66, 66, 66},
Table[n (n + 1)/2, {n, 15}, {n}] // Flatten

王守恩

三角数——{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, ...}

三角数=n个连续自然数(从0开始)的和, 把n个连续自然数两两之差(大数-小数)都加起来, 和可以是这样一串数。
{0, 1, 4, 10, 22, 40, 65, 100, 146, 203, 273, 360, 462, 580, 720, 880, 1060, 1264, 1495, 1750, 2030, 2345, 2688, 3060, 3468, 3912, 4389, 4902, 5460, 6055, 6688, 7370, 8096, 8864, 9680, ...}

把三角数分成自然数的和, 使得这些数两两之差(大数-小数)的积最大。当两两之差(大数-小数)的积最大时, 两两之差(大数-小数)的和可以是这样一串数。
{0, 1, 4, 10, 22, 40, 65, 100, 146, 203, 273, 360, 462, 580, 720, 880, 1060, 1264, 1495, 1750, 2030, 2345, 2688, 3060, 3468, 3912, 4389, 4902, 5460, 6055, 6688, 7370, 8096, 8864, 9680, ...}

OEIS没有这串数。