数字串的通项公式

王守恩

王守恩 发表于 2022-5-29 08:26
2, 3, 5, 8, 12, 19, 30, 48, 76, 121, 192, 305, 484, 768, 1219, 1935, 3072, 4876, 7741, 12288,
  ...

{1, 2, 4, 8, 16, 30, 57, 88, 163, 230, 386, 456, 794, 966, 1471, 1712, 2517, 2484, 4048, 4520, 6196, 6842, 9109, 9048, 12951, 14014,
17902, 19208, 24158, 21510, 31931, 33888, 41449, 43826, 52956, 52992, 66712, 70034, 82993, 86840, 102091, 97776, 124314, 129448}

\(\D a(n)=Mod[n,2]+n\bigg(\frac{n^3-6n^2+23n-18\ \ \ }{24}-\frac{(5n^2-42n+40)del[2,n]\ \ \ }{48}-\frac{3del[4,n]\ }{4}-\frac{(53n-310)del[6,n]\ \ }{12}+\frac{49del[12,n]\ }{2}+32del[18,n]+19del[24,n]-36del[30,n]-50del[42,n]-190del[60,n]-78del[84,n]-48del[90,n]-78del[120,n]-48del[210,n]\bigg)\)

我就好奇：这串数不一定都是在长大。

王守恩

王守恩 发表于 2022-6-4 06:09
{1, 2, 4, 8, 16, 30, 57, 88, 163, 230, 386, 456, 794, 966, 1471, 1712, 2517, 2484, 4048, 4520, 619 ...

自然数 P 次方前 n 项求和公式系数表。

Table[k! StirlingS2[n, k], {n, 11}, {k, n}]
{1}
{1, 2}
{1,  6,  6}
{1, 14, 36,  24}
{1, 30, 150, 240,  120}
{1,  62, 540, 1560,  1800,    720}
{1, 126,1806, 8400, 16800,  15120,   5040}
{1, 254, 5796, 40824,126000, 191520,  141120,    40320}
{1, 510, 18150,186480, 834120, 1905120,  2328480,  1451520,    362880}
{1,1022, 55980, 818520, 5103000, 16435440, 29635200, 30240000,  16329600,  3628800}
{1,2046,171006,3498000,29607600,129230640,322494480,479001600,419126400,199584000,39916800}

附：Table[StirlingS2[n, k], {n, 11}, {k, n}]
{1}
{1,  1}
{1,  3, 1}
{1,  7,  6,  1}
{1, 15, 25, 10,  1}
{1, 31, 90,  65,  15,      1}
{1, 63, 301, 350, 140,    21,          1}
{1, 127, 966, 1701, 1050, 266,       28,          1}
{1, 255, 3025, 7770, 6951, 2646,      462,       36,       1}
{1, 511, 9330, 34105, 42525, 22827,   5880,    750,     45, 1}
{1,1023,28501,145750,246730,179487,63987,11880,1155,55, 1}

王守恩

王守恩 发表于 2022-7-23 15:50
自然数 P 次方前 n 项求和公式系数表。

Table[k! StirlingS2[n, k], {n, 11}, {k, n}]

从简单算起。

\(\D\frac{1}{1^2}>\frac{\pi^2}{6}-\frac{1}{1}\)

\(\D\frac{1}{1^2}+\frac{1}{2^2}>\frac{\pi^2}{6}-\frac{1}{2}>\frac{1}{1^2}\)

\(\D\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}>\frac{\pi^2}{6}-\frac{1}{3}>\frac{1}{1^2}+\frac{1}{2^2}\)

\(\D\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}>\frac{\pi^2}{6}-\frac{1}{4}>\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}\)

\(\D\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}>\frac{\pi^2}{6}-\frac{1}{5}>\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}\)

\(\D\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}>\frac{\pi^2}{6}-\frac{1}{6}>\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}\)


我们可以有：\(n=1, 2, 3, 4, 5, 6, 7, 8, 9, ......\)

\(\D a(n)=\bigg\lfloor\bigg(\frac{\pi^2}{6}-\sum_{k=1}^{n}\ \frac{1}{k^2}\bigg)^{-1}\bigg\rfloor=1, 2, 3, 4, 5, 6, 7, 8, 9, ......\)

王守恩

王守恩 发表于 2022-7-24 16:39
从简单算起。

\(\D\frac{1}{1^2}>\frac{\pi^2}{6}-\frac{1}{1}\)

这些数字串可是在《整数序列在线百科全书(OEIS)》找不到的。

Table[Ceiling[n /\[Pi]], {n, 0, 50}]
{0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16}

Table[Ceiling[n /E], {n, 0, 50}]
{0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19}

Table[Ceiling[(2 n )/\[Pi]], {n, 0, 50}]
{0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32}

Table[Ceiling[(2 n )/E], {n, 0, 50}]
{0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 37}

Table[Ceiling[(3 n )/\[Pi]], {n, 0, 50}]
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 48}

Table[Floor[(3 n )/E], {n, 0, 50}]
{0, 1, 2, 3, 4, 5, 6, 7, 8,  9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55}

Table[Floor[(4 n )/\[Pi]], {n, 0, 50}]
{0, 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63}

............

王守恩

王守恩 发表于 2022-7-28 15:22
这些数字串可是在《整数序列在线百科全书(OEIS)》找不到的。

Table[Ceiling[n /\], {n, 0, 50}]

n 个人围成一圈，选出 k 个人，全部选出的人均不相邻的情况有多少种？

k=1: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
k=2: 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189,209, 230, 252,275, 299,  
k=3: 2, 7, 16, 30, 50, 77, 112, 156, 210, 275, 352, 442,546, 665, 800, 952, 1122, 1311, 1520, 1750,
k=4: 2, 9, 25, 55, 105, 182, 294, 450, 660, 935, 1287, 1729,2275, 2940,3740,4692,5814,7125,8645,
k=5: 2, 11, 36, 91, 196, 378, 672, 1122, 1782, 2717, 4004,5733, 8008, 10948, 14688,19380,25194,
k=6: 2, 13, 49, 140, 336, 714, 1386, 2508, 4290, 7007, 11011, 16744, 24752,35700, 50388, 69768,
k=7: 2, 15, 64, 204, 540, 1254,  2640, 5148,  9438, 16445, 27456, 44200,  68952, 104652, 155040,
k=8: 2, 17, 81, 285, 825, 2079, 4719, 9867, 19305,35750, 63206, 107406,176358, 281010,436050,
k=9: 2, 19, 100, 385, 1210, 3289,  8008, 17875, 37180,  72930, 136136,  243542, 419900, 700910,

\(a(n)=\frac{(n - 1 + 2 k) (n - 2 + k)!}{(n - 1)! k!}\)

这些数字串，可是在《整数序列在线百科全书(OEIS)》不一定找得到的。

王守恩

王守恩 发表于 2022-8-3 12:39
n 个人围成一圈，选出 k 个人，全部选出的人均不相邻的情况有多少种？

k=1: 2, 3, 4, 5, 6, 7, 8, 9, ...

一个数若是 3 的倍数，就除以 3，否则就减去 1，问：几次这样操作后 2022 会变成 1 ？

给出 1——81 的答案。 2022 的答案是15。15次这样操作后 2022 会变成 1。

{0, 1, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 5, 4, 5, 6, 3, 4, 5, 4, 5, 6, 5, 6, 7, 3,
4, 5, 4, 5, 6, 5, 6, 7, 4, 5, 6, 5, 6, 7, 6, 7, 8, 5, 6, 7, 6, 7, 8, 7, 8, 9, 4,
5, 6, 5, 6, 7, 6, 7, 8, 5, 6, 7, 6, 7, 8, 7, 8, 9, 6, 7, 8, 7, 8, 9, 8, 9, 10, 4}

Table[Total[IntegerDigits[n, 3]] + Floor[Log[3, n]] - 1, {n, 1, 81}]

王守恩

A000982       

1, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113, 128, 145, 162, 181, 200, 221, 242, 265, 288,
313, 338, 365, 392, 421, 450, 481, 512, 545, 578, 613, 648, 685, 722, 761, 800, 841, 882, 925, 968,
1013, 1058, 1105, 1152, 1201, 1250, 1301, 1352, 1405, 1458, 1513, 1568, 1625, 1682, 1741, ..........

\(a(n)=\frac{n(n+2)+GCD(n,(n+2))}{2}\)


A183859       

1, 3, 5, 9, 13, 17, 23, 29, 35, 43, 51, 59, 69, 79, 89, 101, 113, 125, 139, 153, 167, 183, 199,
215, 233, 251, 269, 289, 309, 329, 351, 373, 395, 419, 443, 467, 493, 519, 545, 573, 601, 629,
659, 689, 719, 751, 783, 815, 849, 883, 917, 953, 989, 1025, 1063, 1101, 1139, 1179, 1219,.....

\(a(n)=\frac{n(n+3)- GCD(n,(n+3))}{3}\)

王守恩

将 0～n（可重复）填入 n×n 格子，下面不能大于上面，左边不能大于右边，有几种填法？

2, 20, 980, 232848, 267227532, 1478619421136, 39405996318420160,
5055160684040254910720, 3120344782196754906063540800, ......

\(\D a(n)=\prod_{i=1}^{n+1}\ \prod_{j=1}^{n+1}\frac{i+j+n}{i+j-1}\)

aimisiyou

王守恩 发表于 2022-10-3 19:03
将 0～n（可重复）填入 n×n 格子，下面不能大于上面，左边不能大于右边，有几种填法？

2, 20, 980, 2 ...

钩子公式。

王守恩

正方形网格里的最大可填数,满足任意2*2网格都有空格

{1, 3, 8, 12, 21, 27, 40, 48, 65, 75, 96, 108, 133, 147, 176, 192, 225, 243, 280, 300, 341, 363,
408, 432, 481, 507, 560, 588, 645, 675, 736, 768, 833, 867, 936, 972, 1045, 1083, 1160, 1200,
1281, 1323, 1408, 1452, 1541, 1587, 1680, 1728, 1825, 1875, 1976, 2028, 2133, 2187, 2296,....

\(a(n)=n^2-\lfloor\frac{n}{2}\rfloor^2\)