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。

王守恩

01,
02, 03,
04, 06, 05,
07, 09, 10, 08,
11, 13, 15, 14, 12,
16, 18, 20, 21, 19, 17,
22, 24, 26, 28, 27, 25, 23,
29, 31, 33, 35, 36, 34, 32, 30,
37, 39, 41, 43, 45, 44, 42, 40, 38,
46, 48, 50, 52, 54, 55, 53, 51, 49, 47,
56, 58, 60, 62, 64, 66, 65, 63, 61, 59, 57,
67, 69, 71, 73, 75, 77, 78, 76, 74, 72, 70, 68,
79, 81, 83, 85, 87, 89, 91, 90, 88, 86, 84, 82, 80,

{1, 2, 3, 4, 6, 5, 7, 9, 10, 8, 11, 13, 15, 14, 12, 16, 18, 20, 21, 19, 17, 22, 24, 26, 28, 27, 25, 23, 29, 31, 33, 35, 36, 34, 32, 30, 37, 39, 41, 43, 45, 44, 42, 40, 38, 46, 48, 50, 52, 54, 55, 53, 51, 49, 47, 56, 58, 60, 62, 64, 66}

Table[n (n - 1)/2 + Min[2 k - 1, 2 (n - k + 1)], {n, 9}, {k, n}] // Flatten

王守恩

2, 8, 28, 85, 285, 857, 2857, 8571, 28571, 85714, 285714, 857142, 2857142, 8571428, 28571428, 85714285, 285714285, 857142857, 2857142857, 8571428571, 28571428571, 85714285714,

首位数=2,8,2,8,2,8,2,8,2,8,2,8,2,8,2,8,..., 末位数=2,8,8,5,5,7,7,1,1,4,4,2,2,8,8,5,5,7,7,1,1,4,4,...

northwolves

$a_n=\lfloor \frac{1}{7} \sqrt{\left(38-2 (-1)^n\right) 10^n}\rfloor$

王守恩

northwolves 发表于 2025-3-28 12:34
$a_n=\lfloor \frac{1}{7} \sqrt{\left(38-2 (-1)^n\right) 10^n}\rfloor$

序列A057945——0, 1, 2, 1, 2, 3, 1, 2, 3, 2, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 2, 1, 2, 3, 2, 3, 4, 2, 3, 1, 2, 3, 2, 3, 4, 2, 3, 4, 1, 2, 3, 2, 3, 4, 2, 3, 4,
第1次出现数码"1"——1。
第1次出现数码"2"——2。
第1次出现数码"3"——5。
第1次出现数码"4"——20。
第1次出现数码"5"——230。
第1次出现数码"6"——？。
第1次出现数码"7"——？。
第1次出现数码"8"——？。
第1次出现数码"9"——？。

序列A057945——有个挺厉害的条码——如下。我就是不知道怎么把这些 “?” 拉出来。

1. A[n_] := With[{k = Floor[Sqrt[8 n + 1]]}, Floor[(k - 1)/2]*Floor[(k + 1)/2]/2];a[n_] := Module[{k = 0, x = n}, While[x > 0, x = x - A[x]; k++]; k];Table[a[n], {n, 10^99, 10^99 + 10^3}]

复制代码

8, 9, 10, 7, 8, 9, 8, 9, 10, 8, 7, 8, 9, 8, 9, 10, 8, 9, 7, 8, 9, 8, 9, 10, 8, 9, 10, 7, 8, 9, 6, 7, 8, 7, 8, 9, 7, 8, 9, 8, 7, 8, 9, 8, 9, 7, 8, 9, 8, 9, 10, 7, 8, 9, 8, 9, 10, 8, 7, 8, 9, 8, 9, 10, 8, 9, 7, 8, 9, 8, 9, 10, 8, 9, 10, 7, 8, 9, 8, 6, 7, 8, 7,

王守恩

northwolves 发表于 2025-3-28 12:34
$a_n=\lfloor \frac{1}{7} \sqrt{\left(38-2 (-1)^n\right) 10^n}\rfloor$

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, ...}

取三角数中的3个数(最多3个, 取1个也可以, 取2个允许重复, 取3个数不允许重复),  和好像不是任意正整数。因为某些数是无解的, 譬如: 5,8,23,33,57,83, ...,再来几个？

王守恩

n²个球，每种颜色n个球(无区别)。有n种颜色(无区别)，放入n个箱子(无区别)，每个箱子n个，有多少种方案？
a(1)=1,
1, {1},
a(2)=2,
1, {11,22},
2, {12,12},
a(3)=10,
1, {111,222,333},
2, {111,223,233},
3, {112,122,333},
4, {112,123,233},
5, {112,133,223},
6, {113,122,233},
7, {113,123,223},
8, {113,133,222},
9, {122,123,133},
10, {123,123,123},

northwolves

A334286

{1, 2, 10, 465, 190131, 848597563, 43025433375905, 26004966055138634525, 194173310204064149748222455, 18434259996904142171888712495703426, 22778257480946919793779826285286813732062310...}