数字串的通项公式

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A000178    这样不也挺好？        \(\D a(n)=\prod_{k=0}^{n}k!\)

{1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000,
265790267296391946810949632000000000, 127313963299399416749559771247411200000000000, 792786697595796795607377086400871488552960000000000000,
69113789582492712943486800506462734562847413501952000000000000000, 90378331112371142262979521568630736335023247731599748366336000000000000000000,
1890966832292234727042877370627225068196418587883634153182519380410368000000000000000000000}

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northwolves 发表于 2023-12-12 16:44
第一次接触到这个问题是好多好多年前了，大概初中二年级吧。有位邻居小哥哥（曾参加过80年代希望杯数学竞赛 ...

   A226053     这通项公式可以调整吗？
1, 3, 24, 815, 2263886, 9073564639850, 176228569027146222763928594, 84205747605016031994416006285857418872429042805656089,
18266661981464368900241497883663389900558206941880190662422226631656613685596278758397778714969996297024565,

1. Table[Solve[{Floor[-1/x] == x1, Floor[1/(x + x1^-1)] == x2, Floor[-1/(x + x1^-1 - x2^-1)] == x3, Floor[1/(x + x1^-1 - x2^-1 + x3^-1)] == x4,
   
2. Floor[-1/(x + x1^-1 - x2^-1 + x3^-1 - x4^-1)] == x5,Floor[1/(x + x1^-1 - x2^-1 + x3^-1 - x4^-1 + x5^-1)] == x6, Floor[-1/(x + x1^-1 - x2^-1 + x3^-1 - x4^-1 + x5^-1 - x6^-1)] == x7,
   
3. Floor[1/(x + x1^-1 - x2^-1 + x3^-1 - x4^-1 + x5^-1 - x6^-1 + x7^-1)] == x8, Floor[-1/(x + x1^-1 - x2^-1 + x3^-1 - x4^-1 + x5^-1 - x6^-1 + x7^-1 - x8^-1)] == x9,
   
4. Floor[1/(x + x1^-1 - x2^-1 + x3^-1 - x4^-1 + x5^-1 - x6^-1 + x7^-1 - x8^-1 + x9^-1)] == x10}, {x1, x2, x3, x4, x5, x6, x7, x8, x9, x10}], {x, 1/Sqrt[2] - 1, 1/Sqrt[2] - 1}]

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A081721---1, 3, 10, 55, 377, 4291, 60028, 1058058, 21552969, 500280022, 12969598086, 371514016094, 11649073935505, 396857785692525, 14596464294191704, 576460770691256356,

\(\D a(n)=\sum_{k=1}^{n}\frac{n^{GCD(n, k)}}{2n}+\frac{n^{\lceil n/2\rceil}+n^{\lceil(n+1)/2\rceil}}{4}\)

OEIS没有我们的好。还可以更好吗？谢谢！

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A004041        这样不也挺好！？\(\D a(n)=\sum_{k=1}^{n}\frac{(2n-1)!!}{2k-1}\)

1, 4, 23, 176, 1689, 19524, 264207, 4098240, 71697105, 1396704420, 29985521895, 703416314160, 17901641997225, 491250187505700, 14459713484342175, 454441401368236800, ......

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A001175        F周期数       
1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, 16, 30, 48, 24, 100, 84, 72, 48, 14, 120, 30, 48, 40, 36, 80, 24, 76, 18, 56, 60, 40, 48, 88, 30, 120,
48, 32, 24, 112, 300, 72, 84, 108, 72, 20, 48, 72, 42, 58, 120, 60, 30, 48, 96, 140, 120, 136, 36, 48, 240, 70, 24, 148, 228, 200, 18, 80, 168, 78, 120, 216, 120, 168, 48, 180,

1. F周期数公式。Module[{nn = 1000, F}, F = Fibonacci[Range[nn]]; Table[Length[FindTransientRepeat[Mod[F, n], 2][[2]]], {n, 85}]]

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A106291        L周期数       
1, 3, 8, 6, 4, 24, 16, 12, 24, 12, 10, 24, 28, 48, 8, 24, 36, 24, 18, 12, 16, 30, 48, 24, 20, 84, 72, 48, 14, 24, 30, 48, 40, 36, 16, 24, 76, 18, 56, 12, 40, 48, 88, 30, 24,
48, 32, 24, 112, 60, 72, 84, 108, 72, 20, 48, 72, 42, 58, 24, 60, 30, 48, 96, 28, 120, 136, 36, 48, 48, 70, 24, 148, 228, 40, 18, 80, 168, 78, 24, 216, 120, 168, 48, 36,

1. L周期数公式。n = 2; Table[p = i; a = Join[Table[-1, {n - 1}], {n}]; a = Mod[a, p]; b = a; k = 0; While[k++; a = RotateLeft[a]; a[[n]] = Mod[Plus @@ a, p]; b != a]; k, {i, 85}]

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1. N周期数公式。n = 2; Table[p = i; a = Join[Table[-1, {n - 1}], {n}]; a = Mod[a, p]; b = a; k = 0; While[k++; a[[n]] = Mod[Plus @@ a, p]; a = RotateLeft[a]; b != a]; k, {i, 85}]

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1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, 16, 30, 48, 24, 100, 84, 72, 48, 14, 120, 30, 48, 40, 36, 80, 24, 76, 18, 56, 60, 40, 48, 88, 30, 120,
48, 32, 24, 112, 300, 72, 84, 108, 72, 20, 48, 72, 42, 58, 120, 60, 30, 48, 96, 140, 120, 136, 36, 48, 240, 70, 24, 148, 228, 200, 18, 80, 168, 78, 120, 216, 120, 168, 48, 180,

1. V周期数公式。Table[a = {1, 1}; b = a; k = 0; While[k++; a = RotateLeft[a]; a[[2]] = Mod[Plus @@ a, n]; b != a]; k, {n, 2, 303}]

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1. W周期数公式。Table[a = {1, 1}; b = a; k = 0; While[k++; a[[2]] = Mod[Plus @@ a, n]; a = RotateLeft[a]; b != a]; k, {n, 2, 303}]

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{3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, 16, 30, 48, 24, 100, 84, 72, 48, 14, 120, 30, 48, 40, 36, 80, 24, 76, 18, 56, 60, 40, 48, 88, 30, 120,
48, 32, 24, 112, 300, 72, 84, 108, 72, 20, 48, 72, 42, 58, 120, 60, 30, 48, 96, 140, 120, 136, 36, 48, 240, 70, 24, 148, 228, 200, 18, 80, 168, 78, 120, 216, 120, 168, 48, 180,
说明。
1，n=150时, nn=1200, F周期数公式速度慢了。
2，N周期数公式=F周期数公式。一下子我还是找不到反例。各位网友！你能举出反例来吗？谢谢！
3，L周期数公式,N周期数公式。这两者怎么就变了？是怎么变的？这是关键。谢谢！
4，V周期数公式=W周期数公式=F周期数公式。
5，V周期数公式,W周期数公式。这两者怎么就变不出L周期数公式来了？。谢谢！

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1. Table[Sum[ 2 (2 n)! (-1)^(k + 1)/(k (k + 2 n)), {k, \[Infinity]}], {n, 1, 19}]

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1,7,148,6396,468576,52148160,8203541760,1733641056000,473875121664000,162705528979660800,68557495081291776000,34783759238448439296000,
20917982343202389688320000,  14713230137865692249456640000,  11967468761587723030592225280000,  11146324252014844359615538790400000,
11785996694653187027881492375142400000,  14041737888806159459892732515844096000000,  18723173464036641306759645997849116672000000,
......
这串数还可以有其他通项吗？

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A240926      这个通项也可以。
               
4, 5, 9, 20, 49, 125, 324, 845, 2209, 5780, 15129, 39605, 103684, 271445, 710649, 1860500, 4870849, 12752045, 33385284, 87403805,
228826129, 599074580, 1568397609, 4106118245, 10749957124, 28143753125, 73681302249, 192900153620, 505019158609,  ......

1. Table[4*Cosh[n*ArcCsch[2]]^2,{n,0,30}]//FullSimplify

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可以有通项？
0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 103, 140, 188, 242, 327, 429, 568, 755, 1010, 1322, 1751, 2331, 3058, 4067, 5373, 7095, 9409, 12422, 16443, 21754, 28758, 38059,
50328, 66572, 88065, 116471, 154070, 203939, 269699, 356716, 472070, 624281, 825926, 1092650, 1445214, 1911820, 2529171, 3345650, 4425543, 5854772, 7744558, 10244847, 13552871,
17927528, 23716045, 31372514, 41500503, 54899706, 72622993, 96069471, 127085458, 168113818, 222389771, 294187141, 389162566, 514804036, 681006890, 900863709, 1191708300,
1576444545, 2085390562, 2758656887, 3649274077, 4827426837, 6385945524, 8447622535, 11174899003, 14782676306, 19555205716, 25868514369, 34220079641, ......

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0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 103, 140, 188, 242, 327, 429, 568, 755, 1010, 1322, 1751, 2331, 3058, 4067, 5373, 7095, 9409, 12422, 16443, 21754, 28758, 38059,
50328, 66572, 88065, 116471, 154070, 203939, 269699, 356716, 472070, 624281, 825926, 1092650, 1445214, 1911820, 2529171, 3345650, 4425543, 5854772, 7744558, 10244847, 13552871,
17927528, 23716045, 31372514, 41500503, 54899706, 72622993, 96069471, 127085458, 168113818, 222389771, 294187141, 389162566, 514804036, 681006890, 900863709, 1191708300,
1576444545, 2085390562, 2758656887, 3649274077, 4827426837, 6385945524, 8447622535, 11174899003, 14782676306, 19555205716, 25868514369, 34220079641, ......

1. LinearRecurrence[{1,-1,2,-2,3,-3,4,-4,5,-5,6,-6,7,-7,8,-8,9,-9,10,-10,11,-11,12,-12,13,-13},{0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,103,140,188,242,327,429,568},90]

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简单的一串数。OEIS没有。

{-1, 1, 3, 5, 9, 15, 21, 27, 35, 45, 55, 65, 77, 91, 105, 119, 135, 153, 171, 189, 209, 231, 253, 275, 299, 325, 351, 377, 405, 435, 465, 495, 527, 561, 595, 629,...

1. Table[(2 a - 1) (2 b + 3), {a, 9}, {b, 2 a - 4, 2 a - 1}] // Flatten

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求助： -1 可以不出现吗？