3边形数——{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, 561, 595, 630, 666, 703, 741, 780, 820},
4边形数——{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600},
5边形数——{1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380},
6边形数——{1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278, 2415, 2556, 2701, 2850, 3003, 3160},
7边形数——{1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, 3553, 3744, 3940},
8边形数——{1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, 2640, 2821, 3008, 3201, 3400, 3605, 3816, 4033, 4256, 4485, 4720},
9边形数——{1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500}}
A081422——这些数可以由一个公式出来——Table, {n, 3, 9}]——Table, {n, 3, 9}]——Table, {n, 3, 9}]
A083920——Number of nontriangular numbers <= n. 。—— a(n) = n - Round] ——还找得到比这通项公式更简单的吗？！！！
0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57,
{0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 100, ......}
A056847——重复的数是 1, 4, 9, 16, 25, ...——a(n) = n - Round]。
{0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 91, 92, 93, 94, 95, 96, ......}
A083920——重复的数是 1, 3, 6, 10, 15, ...——a(n) = n - Round]。
{0, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62,63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, ......}
OEIS没有——重复的数是 1, 2, 3, 5, 8, 13, 21, 34, ...——a(n) = ？？？？
[欣赏] 下面的每个数字串(k)都可以跑遍所有自然数！——不重不漏, 又不周期循环。
数字串(0) = {0, 2, 1, 5, 4, 3, 9, 8, 7, 6, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 27, 26, 25, 24, 23, 22, 21, 35, 34, 33, 32, 31, 30, 29, 28, 44, 43, 42, 41, 40, 39, 38, 37, 36, 54, 53, 52},
数字串(1) = {1, 0, 5, 4, 3, 2, 11, 10, 9, 8, 7, 6, 19, 18, 17, 16, 15, 14, 13, 12, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 55, 54, 53, 52, 51, 50},
数字串(2) = {2, 1, 0, 8, 7, 6, 5, 4, 3, 17, 16, 15, 14, 13, 12, 11, 10, 9, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 62, 61, 60},
数字串(3) = {3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 59, 58, 57, 56, 55, 54, 53, 52},
数字串(4) = {4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32},
数字串(5) = {5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48},
数字串(6) = {6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 69, 68, 67, 66, 65, 64},
数字串(7) = {7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24},
数字串(8) = {8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33},
数字串(9) = {9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42},
......
A061579——{0, 2, 1, 5, 4, 3, 9, 8, 7, 6, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 27, 26, 25, 24, 23, 22, 21, 35, 34, 33, 32, 31, 30, 29, 28, 44, 43, 42, 41, 40, 39, 38, 37, 36, 54, 53, 52},
A061579——有个数字串(0)——可通项公式没我们的好——a(n) = Round]^2 - n 。
其它的数字串(k)——OEIS就没有了。
上面的数字串可以有一个统一的通项公式——a(k) = k*Round]^2 - n 。—— 精妙在于:"2"不能改。 "1"不能少。 动"1/4"就漏解。不精妙的就不显臭了。
OEIS没有这数字串。
{0, 1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 49, 50, 51, 52, 53, 54, 55, 56}
Flatten, {k, 9}]][[;; 99]]
Table]^2, {n, 99}]
OEIS没有这串数。看5个公式之间的差异——备忘。
{1, 0, 3, 2, 6, 5, 4, 9, 8, 7, 13, 12, 11, 10, 17, 16, 15, 14, 22, 21, 20, 19, 18, 27, 26, 25, 24, 23, 33, 32, 31, 30, 29, 28, 39, 38, 37, 36, 35, 34, 46, 45, 44, 43, 42, 41, 40, 53, 52, 51, 50, 49, 48, 47, 61, 60, 59, 58, 57,
56, 55, 54, 69, 68, 67, 66, 65, 64, 63, 62, 78, 77, 76, 75, 74, 73, 72, 71, 70, 87, 86, 85, 84, 83, 82, 81, 80, 79, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 118, 117, 116, ...}
M = 20; s = {1}; t = {2}; d = 2; For, d = d + 1]]; AppendTo] + d]; AppendTo, t[] + 1, t[]]];]
f = {}; For], s[] - t[] + 1, -1]];]Take
s = 1; t = 2; d = 2; f = {1, 0}; Do; s += d; t += Boole@OddQ@i; f = Join], {i, 2, 20}]; Take
Flatten] + Floor[(n^2 + 4 n - 8)/4], {n, 20}]][[;; 120]]
Flatten] + Floor[(n + 2)^2/4] - 3, {n, 20}]][[;; 120]]
Flatten] + Floor[(n/2)^2] - 3, {n, 3, 22}]][[;; 120]]
A219233——a(n) = (-1)^n*LucasL - Boole—— (* G. C. Greubel, Jun 13 2025 *)
1, -3, 7, -18, 47, -123, 322, -843, 2207, -5778, 15127, -39603, 103682, -271443, 710647, -1860498, 4870847, -12752043, 33385282, -87403803, 228826127, -599074578, 1568397607, -4106118243, 10749957122, -28143753123,
a(n)=\(\bigg[\big(\frac{3 + \sqrt{5}}{2 \cos(n*\pi)}\big)^n\bigg]\)——这个简单些！
{1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62,63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100}
OEIS没有——重复的数是 1, 2, 3, 5, 8, 13, 21, 34, ...——a(n) = ？？？？
Table, _?(# >= k &)][]}, Fibonacci + (k - If[]] - 1)], {k, 1, 110}]
Table)^(m + 1) - (1 - Sqrt)^(m + 1))/(Sqrt*2^(m + 1))] + m - 1, {m, 10}], _?(# >= k &)][]},
Fibonacci + (k - If)^n - (1 - Sqrt)^n)/(Sqrt*2^n)] + n - 2] - 1)], {k, 1, 110}]
w = Table5)/2)^(m + 1)/\5] + m - 1, {m, 11}]; Table[(n = FirstPosition[]; Fibonacci@n + k - If] + 1]), {k, 110}]
Table, Table)^k/(2^k Sqrt)], {k, 11}]]], {n, 100, 100}]
Table, Table, {k, 11}]]], {n, 100, 100}]
Flatten, Fibonacci}]][] & /@ {110}
Flatten, Fibonacci}]][[;; 110]]
7 个代码显示——上面同一个数字串 1——100。
A^a(n) is the smallest power of A beginning with n.——来个通吃公式——OEIS没我们简捷！
{0, 1, 5, 2, 9, 6, 46, 3, 53, 10, 50, 7, 17, 47, 77, 4, 34, 54, 84, 11, 31, 51, 61, 81, 8, 18, 38, 48, 68, 78, 98, 5, 25, 35, 45, 55, 75, 85, 95, 12}, ——A018856——Aug 31 2025
{0, 3, 1, 14, 10, 8, 6, 4, 2, 21, 19, 17, 38, 15, 13, 34, 11, 32, 9, 30, 7, 28, 49, 5, 26, 47, 3, 24, 45, 66, 22, 43, 108, 20, 85, 41, 106, 18, 83, 39}, ——A018858
{0, 4, 39, 1, 51, 3, 23, 53, 78, 5, 25, 40, 60, 75, 90, 2, 17, 27, 42, 52, 62, 72, 82, 92, 4, 9, 19, 24, 34, 39, 49, 54, 59, 64, 74, 79, 84, 89, 94, 6}, ——A018860
{0, 2, 5, 11, 1, 4, 7, 50, 10, 63, 23, 3, 56, 26, 6, 69, 49, 29, 9, 82, 62, 42, 22, 12, 2, 75, 55, 45, 25, 15, 5, 88, 68, 58, 48, 38, 28, 18, 8, 91}, ——A018862——Aug 19 2025
{0, 3, 2, 6, 87, 1, 5, 41, 176, 9, 117, 4, 13, 94, 175, 8, 26, 89, 152, 215, 3, 30, 84, 129, 174, 219, 7, 16, 52, 88, 133, 169, 205, 241, 268, 2, 38, 65, 101, 128},
{0, 4, 3, 2, 8, 14, 1, 7, 13, 45, 6, 38, 12, 57, 31, 5, 50, 37, 11, 69, 43, 30, 17, 4, 49, 36, 23, 10, 68, 55, 42, 29, 16, 3, 61, 48, 35, 106, 22, 9}, ——A018866——Feb 22 2018
{0, 6, 5, 4, 3, 2, 22, 1, 21, 10, 20, 30, 9, 19, 29, 8, 49, 18, 28, 7, 110, 17, 58, 27, 37, 6, 47, 16, 57, 26, 98, 5, 139, 46, 15, 118, 25, 190, 97, 4}, ——A018868
{0, 12, 9, 7, 5, 4, 3, 2, 1, 21, 42, 20, 19, 40, 18, 17, 60, 16, 59, 15, 80, 14, 101, 57, 13, 78, 34, 12, 77, 33, 11, 98, 54, 10, 119, 75, 53, 9, 118, 96},——A018870——Dec 09 2021
......
Table, k = 0}, While, UpTo]] != t, k++]; k], {A, 2, 9}, {n, 40}]
A138173——a(n) is the smallest m such that k^3 begins with n^2.—— May 24 2016。
1, 16, 21, 55, 63, 154, 17, 4, 201, 10, 23, 113, 257, 27, 609, 295, 307, 148, 1535, 342, 164, 1692, 809, 1793, 397, 878, 9, 428, 944, 4482, 987, 1008, 1029, 4872, 107, 2349, 5154, 5247, 2478, 252, 552, 5609, 5697, 5785, 2726,
Table, k = 0}, While, UpTo]] != t, k++]; k], {n, 48}]——我来补个公式。