数字串的通项公式

northwolves

王守恩 发表于 2023-3-20 17:51
谢谢 northwolves！     A038597

{0, 13, 28, 49,  104, 147, 181, 189, 224, 351, 361, 388, 39 ...

1. Select[Range@620,
   
2. AnyTrue[Table[CubeRoot[#^2 + a^3], {a, #}], IntegerQ] &]

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{13, 28, 49, 104, 147, 181, 189, 224, 351, 361, 388, 392, 507, 549, 588}

王守恩

谢谢northwolves！我缺的就是这些 “方法”。谢谢northwolves！

{0, 13, 28, 49,  104, 147, 181, 189, 224, 351, 361, 388, 392, 507, 549,  588,......
算式(1): Select[Union@Flatten@Table[Sqrt[b^3 - a^3], {a, 1, 6200}, {b, a, Power[a^3 + 6200^2, (3)^-1]}], IntegerQ[#] &]

{13, 28, 49, 104, 147, 181, 189, 224, 351, 361, 388, 392, 507, 549, 588,......
算式(2): Select[Range@6200, AnyTrue[Table[CubeRoot[#^2 + a^3], {a, #}], IntegerQ] &]

算式(1)多了个0(能去掉吗?)，但比算式(2)出答案的速度好像快一些。

王守恩

A173196       
       
{0, 1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715,825, 946, 1078, 1222, 1378, 1547,
1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843,
8372, 8924, 9500, 10100, 10725,  11375, 12051, 12753, 13482, 14238,  15022, 15834, 16675, 17545,  18445, ......... }

\(a(n)=\D\sum_{k=1}^n\bigg\lfloor\frac{k^2}{4}\bigg\rfloor\)

王守恩

A005408     ContinuedFraction[(E^2 + 1)/(E^2 - 1), 41]
       
1,  3,  5,  7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49,
51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97,
99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, ......

白新岭

王守恩 发表于 2023-4-13 15:13
A005408     ContinuedFraction[(E^2 + 1)/(E^2 - 1), 41]
       
1,  3,  5,  7, 9, 11, 13, 15, 17, 19, 21 ...

王守恩玩数字串很溜达，我是玩不来，我能玩的就是错位相加法！

王守恩

A028387                \(a(n) =\sqrt{(n-1)*n*(n + 1)*(n + 2) + 1}\)

{1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461,505, 551,
599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639,1721,
1805, 1891, 1979, 2069, 2161, 2255, 2351, 2449, 2549, 2651, 2755, 2861, 2969, 3079, 3191, ......

王守恩

题目:投篮测试考核,若连续n次投篮命中,则测试成功,若连续k次投篮失败,则测试失败

投篮命中率=\(\frac{A}{A+1}\),且k趋向无穷大

投篮命中率=1/2,考核要求连续1,2,3,4,......次投篮命中,k趋向无穷大,测试成功率:
{5/6, 11/20, 23/72, 47/272, 95/1056, 191/4160, 383/16512, 767/65792, 1535/262656, 3071/1049600, 6143/4196352, 12287/16781312, 24575/67117056, 49151/268451840},

投篮命中率=2/3,考核要求连续1,2,3,4,......次投篮命中,k趋向无穷大,测试成功率:
{20/21, 128/153, 800/1161, 4928/9153, 30080/74601, 182528/624753, 1103360/5342841, 6652928/46405953, 40048640/407575881, 240816128/3607716753, 1446993920/32106653721,
  
投篮命中率=3/4,考核要求连续1,2,3,4,......次投篮命中,k趋向无穷大,测试成功率:
{51/52, 639/688, 7911/9280, 97119/127744, 1185111/1795072, 14398479/25735168, 174376071/375930880, 2106861759/5584912384, 25411481271/84198817792, 306100036719/1285263720448,

投篮命中率=4/5,考核要求连续1,2,3,4,......次投篮命中,k趋向无穷大,测试成功率:
{104/105, 2144/2225, 43904/47625, 894464/1030625, 18151424/22565625, 367222784/500140625, 7411564544/11223515625, 149305032704/254987890625, 3003280523264/5862697265625,

投篮命中率=5/6,考核要求连续1,2,3,4,......次投篮命中,k趋向无穷大,测试成功率:
{185/186, 5675/5796, 173375/181656, 5279375/5729616, 160334375/181966176, 4858859375/5821782336, 146986484375/187714164096, 4440112109375/6101609907456, 133966302734375/1999749

投篮命中率=6/7,考核要求连续1,2,3,4,......次投篮命中,k趋向无穷大,测试成功率:
{300/301, 12816/12985, 546048/562177, 23213952/24434977, 985063680/1066622641, 41735471616/46775477665, 1765950501888/2061459072337, 74640106063872/91328842548097, 31518111141273
  
投篮命中率=7/8,考核要求连续1,2,3,4,......次投篮命中,k趋向无穷大,测试成功率:
{455/456, 25823/26048, 1462895/1491456, 82745663/85618688, 4674110735/4928864256, 263727527903/284606332928, 14865630572975/16487710457856, 837222873596543/958496157728768,

投篮命中率=8/9,考核要求连续1,2,3,4,......次投篮命中,k趋向无穷大,测试成功率:
{656/657, 47744/48033, 3470336/3517425,251961344/258037569, 18275434496/18966125457, 1324421218304/1396942092513, 95908083531776/103121696489265, 6940566386376704/7630653279321729,   

投篮命中率=9/10,考核要求连续1,2,3,4,......次投篮命中,k趋向无穷大,测试成功率:
{909/910, 82539/82900, 7487559/7561000, 678663279/690490000, 61467115599/63144100000, 5563421463519/5782969000000, 503249797545039/530467210000000, 45498372911148159/48742048900000000,

通项公式:  \(\D\frac{A^n((A+1)^n(A+2)-A^n)}{(A+1)^n(A^{n+1}+(A+1)^n)}\)    这些数字串(分子,分母)可都是在OEIS找不到的。

王守恩

{2, 03, 005, 009, 017, 033, 065, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577,-A000051       
{3, 07, 017, 043, 113, 307, 857, 2443, 7073, 20707, 61097, 181243, 539633, 1610707, 4815737, 14414443, 43177793, 129402307,-A085279       
{4, 13, 043, 145, 499, 1753, 6283, 22945, 85219, 321193, 1225723, 4725745, 18371539, 71891833, 282784363, 1116788545, ----?  
{5, 21, 089, 381, 1649, 7221, 32009, 143661, 652769, 3001701, 13959929, 65605341, 311249489, 1489138581, 7177257449, ----?
{6, 31, 161, 841, 4421, 23401, 124781, 670561, 3632741, 19843321, 109294301, 606937681, 3397485461,19164209641,      -----?
{7, 43, 265, 1639, 10177, 63463, 397585, 2503159,15842497, 100819783, 645272305, 4154109079, 26901981217,            --------?
{8, 57, 407, 2913, 20903, 150417, 1085687, 7861953, 57130823, 416692977, 3051068567, 22431221793, 165608487143,       -----?
{9, 73, 593, 4825, 39329, 321193, 2628593, 21560185, 177264449, 1461162313, 12076718993, 100100536345, 832185350369, ---?


补充内容 (2023-5-13 05:23):
通项公式：Table[(n + 1)^m + n^(m + 1), {n, 1, 9}, {m, 0, 8}]

王守恩

就这么些数字串，可是在《整数序列在线百科全书(OEIS)》找不到的。

{2, 04, 08, 16, 032, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072,
{3, 07, 19, 55, 163, 487, 1459, 4375, 13123, 39367,118099,354295, 1062883, 3188647, 9565939,
{4, 12, 40, 144, 544, 2112, 8320, 33024,131584,525312,2099200,8392704,33562624,134234112,
{5, 19, 77, 331, 1493, 6979, 33437, 162811, 800933, 3965299, 19708397, 98187691, 489875573,
{6, 28, 136, 688, 3616, 19648, 109696, 625408, 3621376, 21203968, 125126656, 742371328,
{7, 39, 223, 1311, 7927, 49239, 313423, 2037711, 13482727, 90472839,613778623,4198794111,
{8, 52, 344, 2320, 15968, 112192, 804224, 5873920, 43632128, 328901632, 2510280704,
{9, 67, 505, 3859, 29929, 235747, 1886425, 15330739, 126447049, 1057316227, 8950895545,

王守恩

若 n={2, 8, 10, 12, 18, 20, 28, 30, 32, 40, 42, 48, 50, 60, 68, 70, 72, 78, 80, 88, 90, 98, 102, 108, ......

则 abc(a+b+c) =2n^2 有正整数解 。 这串数好像在《整数序列在线百科全书(OEIS)》找不到。

1. Select[Union@Flatten@Table[Sqrt[(a*b*c (a + b + c))/2],{a, 1, 20}, {b, a, 20}, {c, b, 32}],  IntegerQ[#] &]

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公式不好: a,b,c 取值浪费太大，a,b,c 应该如何取值？谢谢！