数字串的通项公式

王守恩 · 发表于 2025-2-19 14:04:56

k = 3 ——后面不会出现2——{1, 2, 7, 11, 101, 111, 1001, 1011, 2201, 10001, 10011, 10101, 11011, 100001, 100011, 100101, 100111, 101011, 101101, 110011, 1000001, 1000011, 1000101, 1000111, 1001001,
1001011, 1001101, 1010011, 1100011, 10000001, 10000011, 10000101, 10000111, 10001001, 10001011, 10001101, 10010011, 10010101, 10011001, 10100011, 10100101, 11000011, ......

又: k = 3 有点慢, 我们就规定 x 是 1 与 0 组成的十进制数, 会快一些, 我只能这样编。

NSolve[{10^10 + k*10^9 + j*10^8 + h*10^7 + g*10^6 + f*10^5 + d*10^4 + c*10^3 + b*10^2 + a*10 + 1 == x, IntegerReverse[x^3] == y^3, y >= x,
1 >= k >= 0, 1 >= j >= 0, 1 >= h >= 0, 1 >= g >= 0, 1 >= f >= 0, 1 >= d >= 0, 1 >= c >= 0, 1 >= b >= 0, 1 >= a >= 0}, {x, y, k, j, h, g, f, d, c, b, a}, Integers]

{x -> 10000000001, y -> 10000000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 0, c -> 0, b -> 0, a -> 0},
{x -> 10000000011, y -> 11000000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 0, c -> 0, b -> 0, a -> 1},
{x -> 10000000101, y -> 10100000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 0, c -> 0, b -> 1, a -> 0},
{x -> 10000000111, y -> 11100000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 0, c -> 0, b -> 1, a -> 1},
{x -> 10000001001, y -> 10010000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 0, c -> 1, b -> 0, a -> 0},
{x -> 10000001011, y -> 11010000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 0, c -> 1, b -> 0, a -> 1},
{x -> 10000001101, y -> 10110000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 0, c -> 1, b -> 1, a -> 0},
{x -> 10000010001, y -> 10001000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 1, c -> 0, b -> 0, a -> 0},
{x -> 10000010011, y -> 11001000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 1, c -> 0, b -> 0, a -> 1},
{x -> 10000010101, y -> 10101000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 1, c -> 0, b -> 1, a -> 0},
{x -> 10000011001, y -> 10011000001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 0, d -> 1, c -> 1, b -> 0, a -> 0},
{x -> 10000100001, y -> 10000100001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 1, d -> 0, c -> 0, b -> 0, a -> 0},
{x -> 10000100011, y -> 11000100001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 1, d -> 0, c -> 0, b -> 0, a -> 1},
{x -> 10000100101, y -> 10100100001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 1, d -> 0, c -> 0, b -> 1, a -> 0},
{x -> 10000101001, y -> 10010100001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 1, d -> 0, c -> 1, b -> 0, a -> 0},
{x -> 10000110001, y -> 10001100001, k -> 0, j -> 0, h -> 0, g -> 0, f -> 1, d -> 1, c -> 0, b -> 0, a -> 0},
{x -> 10001000011, y -> 11000010001, k -> 0, j -> 0, h -> 0, g -> 1, f -> 0, d -> 0, c -> 0, b -> 0, a -> 1},
{x -> 10001000101, y -> 10100010001, k -> 0, j -> 0, h -> 0, g -> 1, f -> 0, d -> 0, c -> 0, b -> 1, a -> 0},
{x -> 10001001001, y -> 10010010001, k -> 0, j -> 0, h -> 0, g -> 1, f -> 0, d -> 0, c -> 1, b -> 0, a -> 0},
{x -> 10001001011, y -> 11010010001, k -> 0, j -> 0, h -> 0, g -> 1, f -> 0, d -> 0, c -> 1, b -> 0, a -> 1},
{x -> 10001010001, y -> 10001010001, k -> 0, j -> 0, h -> 0, g -> 1, f -> 0, d -> 1, c -> 0, b -> 0, a -> 0},
{x -> 10010000011, y -> 11000001001, k -> 0, j -> 0, h -> 1, g -> 0, f -> 0, d -> 0, c -> 0, b -> 0, a -> 1},
{x -> 10010000101, y -> 10100001001, k -> 0, j -> 0, h -> 1, g -> 0, f -> 0, d -> 0, c -> 0, b -> 1, a -> 0},
{x -> 10010000111, y -> 11100001001, k -> 0, j -> 0, h -> 1, g -> 0, f -> 0, d -> 0, c -> 0, b -> 1, a -> 1},
{x -> 10010001001, y -> 10010001001, k -> 0, j -> 0, h -> 1, g -> 0, f -> 0, d -> 0, c -> 1, b -> 0, a -> 0},
{x -> 10011000101, y -> 10100011001, k -> 0, j -> 0, h -> 1, g -> 1, f -> 0, d -> 0, c -> 0, b -> 1, a -> 0},
{x -> 10100000011, y -> 11000000101, k -> 0, j -> 1, h -> 0, g -> 0, f -> 0, d -> 0, c -> 0, b -> 0, a -> 1},
{x -> 10100000101, y -> 10100000101, k -> 0, j -> 1, h -> 0, g -> 0, f -> 0, d -> 0, c -> 0, b -> 1, a -> 0},
{x -> 10100100011, y -> 11000100101, k -> 0, j -> 1, h -> 0, g -> 0, f -> 1, d -> 0, c -> 0, b -> 0, a -> 1},
{x -> 10101000011, y -> 11000010101, k -> 0, j -> 1, h -> 0, g -> 1, f -> 0, d -> 0, c -> 0, b -> 0, a -> 1},
{x -> 11000000011, y -> 11000000011, k -> 1, j -> 0, h -> 0, g -> 0, f -> 0, d -> 0, c -> 0, b -> 0, a -> 1}}

northwolves · 发表于 2025-2-19 18:17:07

王守恩发表于 2025-2-19 14:04
k = 3 ——后面不会出现2——{1, 2, 7, 11, 101, 111, 1001, 1011, 2201, 10001, 10011, 10101, 11011, 100 ...

s={};For[k=17,k<2^12,k+=2,x=FromDigits@IntegerDigits[k,2];y=CubeRoot@IntegerReverse[x^3];If[IntegerQ@y && y>=x,AppendTo[s,x]]];s

复制代码

{10001,10011,10101,11011,100001,100011,100101,100111,101011,101101,110011,1000001,1000011,1000101,1000111,1001001,1001011,1001101,1010011,1100011,10000001,10000011,10000101,10000111,10001001,10001011,10001101,10010011,10010101,10011001,10100011,10100101,11000011,100000001,100000011,100000101,100000111,100001001,100001011,100001101,100010001,100010011,100011001,100100011,100100101,100101001,101000011,101000101,110000011,1000000001,1000000011,1000000101,1000000111,1000001001,1000001011,1000001101,1000010001,1000010011,1000010101,1000011001,1000100011,1000100101,1000101001,1000110001,1001000011,1001000101,1001000111,1010000011,1010000101,1010010011,1100000011,10000000001,10000000011,10000000101,10000000111,10000001001,10000001011,10000001101,10000010001,10000010011,10000010101,10000011001,10000100001,10000100011,10000100101,10000101001,10000110001,10001000011,10001000101,10001001001,10001001011,10001010001,10010000011,10010000101,10010000111,10010001001,10011000101,10100000011,10100000101,10100100011,10101000011,11000000011,100000000001,100000000011,100000000101,100000000111,100000001001,100000001011,100000001101,100000010001,100000010011,100000010101,100000011001,100000100001,100000100011,100000100101,100000100111,100000101001,100000101011,100000110001,100000111001,100001000011,100001000101,100001001001,100001001101,100001010001,100001011001,100001100001,100010000011,100010000101,100010000111,100010001001,100010001011,100010001101,100010010001,100100000011,100100000101,100100000111,100100001001,100100100011,100101000011,100110000101,101000000011,101000000101,101000010011,101000100011,101010000011,110000000011}

northwolves · 发表于 2025-2-19 22:32:05

本帖最后由 northwolves 于 2025-2-19 22:33 编辑

northwolves 发表于 2025-2-19 18:17
{10001,10011,10101,11011,100001,100011,100101,100111,101011,101101,110011,1000001,1000011,100010 ...

或者

f[n_] := CubeRoot@IntegerReverse[n^3];
Select[FromDigits@IntegerDigits[#, 2] & /@ Range[17, 2^12, 2], IntegerQ@f@# && f@# >= # &]

复制代码

王守恩 · 发表于 2025-2-20 09:47:43

A000255——1, 1, 3, 11, 53, 309, 2119, 16687, 148329, 1468457, 16019531, 190899411, 2467007773, 34361893981, 513137616783, 8178130767479, 138547156531409, 2486151753313617, 47106033220679059, 939765362752547227, ...
这样不就行了！——Table[Subfactorial[n + 1]/n, {n, 35}]

王守恩 · 发表于 2025-2-21 18:50:27

A080637——规律: a(2n+1) = 2*a(n) + 1, a(2n) = a(n) + a(n-1) + 1.

A080637——{2, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85,87,
89,91, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 129, 131, 133, 135, 137, 139, 141, 143,145, 147, 149, 151,
153,155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217,
218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 257, 259, 261, 263, 265, 267, 269, 271,
273, 275, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 301, 303, 305, 307, 309, 311, 313, 315, 317, 319, 321, 323, 325, 327, 329, 331, 333, 335, 337, 339, 341, 343, 345, 347, 349, 351, 353, 355, 357, 359, 361, 363,
365, 367, 369, 371, 373, 375, 377, 379, 381, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419,
420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465,
466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511,

公式(1)——b[n_] := b[n] = If[n < 4, n + 1, If[OddQ[n],2 b[n/2], b[(n + 1)/2] + b[(n - 1)/2]]]; a[n_] := b[n + 1] - 1; a /@ Range[70]

A171757——{4, 8, 10, 16, 18, 20, 22, 32, 34, 36, 38, 40, 42, 44, 46, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158,
160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 256, 258, 260, 262, 264, 266, 268, 270, 272, 274, 276, 278, 280, 282, 284, 286, 288, 290, 292, 294, 296, 298, 300, 302, 304, 306,
308, 310, 312, 314, 316, 318, 320, 322, 324, 326, 328, 330, 332, 334, 336, 338, 340, 342, 344, 346, 348, 350, 352, 354, 356, 358, 360, 362, 364, 366, 368, 370, 372, 374, 376, 378, 380, 382}

公式(2)——2 Select[Range[2, 180], # <= 2 || Take[IntegerDigits[#, 2], 2] != {1, 1} &]

公式(3)——RecurrenceTable[{b[1] == 2, b[2] == 3, b[n] == b[Floor[n/2]] + b[Floor[(n - 1)/2]]}, a, {n, 70}]

2个问题。

1，A171757是A080637的补集。可以用公式(2)把A080637拉出来吗？

2，公式(3)应该=公式(1)，可公式(3)就是怎么也拉不出数字串来？那里出问题了？

王守恩 · 发表于 2025-2-23 07:50:44

我还是纠结楼上的疑惑。类似的(OEIS找不到)通项公式怎么来？

0+0+1=1
0+1+1=2
0+1+1=3
1+1+2=4
1+2+2=5
2+2+2=6
2+2+3=7
2+3+3=8
3+3+3=9
3+3+5=11
3+5+5=13
5+5+5=15
5+5+8=18
5+8+8=21
8+8+8=24
8+8+13=29
8+13+13=34
13+13+13=39
13+13+21=47
13+21+21=55
21+21+21=63
21+21+34=76
21+34+34=89
34+34+34=102

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 18, 21, 24, 29, 34, 39, 47, 55, 63, 76, 89, 102, 123, 144, 165, 199, 233, 267, 322, 377, 432, 521, 610, 699, 843, 987, 1131, 1364, 1597, 1830,

王守恩 · 发表于 2025-2-25 13:05:43

LinearRecurrence[{1, 0, 1}, {1, 1, 2}, 70]

{1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, ...

1+2=3,
1+3=4,
2+4=6,
3+6=9,
4+9=13,
6+13=19,
9+19=28,

就这么简单！也是不可以有通项公式的！

Table[RootSum[1 + #^2 - #^3 &, -3 #^(n + 2) + 11 #^(n + 3) - 2 #^(n + 4) &]/31, {n, 70}]—— 2025 年 2 月 14 日。

{1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, ...

A000930——没看懂——备忘一下。

northwolves · 发表于 2025-2-25 15:41:50

王守恩发表于 2025-2-25 13:05
LinearRecurrence[{1, 0, 1}, {1, 1, 2}, 70]

{1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, ...

$a_n=\text{Round}\left[\left(\root[3]{\frac{4}{837}+\frac{4}{\sqrt{93^3}}}+\root[3]{\frac{4}{837}-\frac{4}{\sqrt{93^3}}}+\frac{1}{3}\right) \left(\root[3]{\frac{29}{54}}+\frac{\sqrt{93}}{18}+\root[3]{\frac{29}{54}-\frac{\sqrt{93}}{18}}+\frac{1}{3}\right)^n\right]$

northwolves · 发表于 2025-2-25 15:51:42

或者OEIS提供的公式：

$a_n=\lfloor\left(\root[3]{\frac{4}{837}+\frac{4}{\sqrt{93^3}}}+\root[3]{\frac{4}{837}-\frac{4}{\sqrt{93^3}}}+\frac{1}{3}\right) \left(\root[3]{\frac{29}{54}}+\frac{\sqrt{93}}{18}+\root[3]{\frac{29}{54}-\frac{\sqrt{93}}{18}}+\frac{1}{3}\right)^n+\frac{1}{2} \rfloor$

王守恩 · 发表于 2025-2-25 18:41:36

northwolves 发表于 2025-2-25 15:41
$a_n=\text{Round}\left[\left(\root[3]{\frac{4}{837}+\frac{4}{\sqrt{93^3}}}+\root[3]{\frac{4}{837}- ...

Table[Round[(Power[4/837 + 4/Sqrt[93^3], (3)^-1] + Power[4/837 - 4/Sqrt[93^3], (3)^-1] + 1/3) (Power[29/54 + Sqrt[93/18^2], (3)^-1] + Power[29/54 - Sqrt[93/18^2], (3)^-1] + 1/3)^n], {n, 40}]
{1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964}
Table[HypergeometricPFQ[{(2 - n)/3, (1 - n)/3, (0 - n)/3}, {(1 - n)/2, (0 - n)/2}, 3^3/-2^2], {n, 40}]
{1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964}
Table[HypergeometricPFQ[{(3 - n)/4, (2 - n)/4, (1 - n)/4, (0 - n)/4}, {(2 - n)/3, (1 - n)/3, (0 - n)/3}, 4^4/-3^3], {n, 50}]
{1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 95, 131, 181, 250, 345, 476, 657, 907, 1252, 1728, 2385, 3292, 4544, 6272, 8657, 11949, 16493, 22765, 31422, 43371, 59864, 82629, 114051, 157422, 217286, 299915——A003269。
Table[HypergeometricPFQ[{(4 - n)/5, (3 - n)/5, (2 - n)/5, (1 - n)/5, (0 - n)/5}, {(3 - n)/4, (2 - n)/4, (1 - n)/4, (0 - n)/4}, 5^5/-4^4], {n, 50}]
{1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 106, 140, 185, 245, 325, 431, 571, 756, 1001, 1326, 1757, 2328, 3084, 4085, 5411, 7168, 9496, 12580, 16665, 22076, 29244, 38740, 51320, 67985, 90061——A003520。
Table[HypergeometricPFQ[{(5 - n)/6, (4 - n)/6, (3 - n)/6, (2 - n)/6, (1 - n)/6, (0 - n)/6}, {(4 - n)/5, (3 - n)/5, (2 - n)/5, (1 - n)/5, (0 - n)/5}, 6^6/-5^5], {n, 50}]
{1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 43, 55, 71, 92, 119, 153, 196, 251, 322, 414, 533, 686, 882, 1133, 1455, 1869, 2402, 3088, 3970, 5103, 6558, 8427, 10829, 13917, 17887, 22990, 29548——A005708。
Table[HypergeometricPFQ[{(3 - n)/4, (2 - n)/4, (1 - n)/4, (0 - n)/4}, {(1 - n)/2, (0 - n)/2}, 4^4/2^2], {n, 50}]
{1, 1, 1, 3, 7, 13, 21, 43, 103, 237, 493, 1051, 2463, 6013, 14197, 33003, 79351, 198733, 499773, 1244347, 3124303, 8038941, 20991493, 54843403, 143628327, 380955373, 1024406221, 2772060123——OEIS没有了。
Table[HypergeometricPFQ[{(4 - n)/5, (3 - n)/5, (2 - n)/5, (1 - n)/5, (0 - n)/5}, {(1 - n)/2, (0 - n)/2}, 5^5/-2^2], {n, 50}]
{1, 1, 1, 1, 7, 25, 61, 121, 211, 697, 3025, 10801, 31231, 76921, 228517, 939625, 3946891, 14390041, 45049081, 143927137, 566622775, 2514795481, 10561116301, 39278788441, 140006432707——OEIS没有了。
Table[HypergeometricPFQ[{(4 - n)/5, (3 - n)/5, (2 - n)/5, (1 - n)/5, (0 - n)/5}, {(2 - n)/3, (1 - n)/3, (0 - n)/3}, 5^5/3^3], {n, 50}]
{1, 1, 1, 1, 3, 7, 13, 21, 31, 55, 117, 253, 511, 951, 1765, 3517, 7503, 16231, 34021, 69405, 142687, 303703, 666501, 1469341, 3200335, 6921447, 15106693, 33614461, 75869151, 171629335——OEIS没有了。

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[原创] 数字串的通项公式

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