求通项公式

王守恩

求通项公式: 12, 127, 1277, 12777, 127777, 1277777, 12777777, 127777777, ...

Nicolas2050

你这辈子也就搞这了

ysr

王守恩 发表于 2023-4-29 04:24
求通项公式: 12, 127, 1277, 12777, 127777, 1277777, 12777777, 127777777, ...

12*10^(n-1)+(1-1/10^(n-1))*7*10^(n-1)/9，其中的n大于等于1

王守恩

ysr 发表于 2023-4-29 22:05
12*10^(n-1)+(1-1/10^(n-1))*7*10^(n-1)/9，其中的n大于等于1

\(12*10^{n-1}+(1-\frac{1}{10^{n-1}})*7*\frac{10^{n-1}}{9}\)

=\(12*10^{n}+(1-\frac{1}{10^{n}})*7*\frac{10^{n}}{9}\)

=\(\frac{(12*9*10^{n}+7*10^n-7)*10^{n}}{9*10^{n}}\)

=\(\frac{115*10^{n}-7}{9}\)

=\(\big\lfloor\frac{115*10^{n}}{9}\big\rfloor\)

=\(\big\lfloor\frac{23*10^{n}}{18}\big\rfloor\)

=\(\frac{23*10^{n}-14}{18}\)

王守恩

求通项公式
{2, 03, 005, 009, 017, 033, 065, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537,         
{3, 07, 017, 043, 113, 307, 857, 2443, 7073, 20707, 61097, 181243, 539633, 1610707, 4815737,        
{4, 13, 043, 145, 499, 1753, 6283, 22945, 85219, 321193, 1225723, 4725745, 18371539,
{5, 21, 089, 381, 1649, 7221, 32009, 143661, 652769, 3001701, 13959929, 65605341,
{6, 31, 161, 841, 4421, 23401, 124781, 670561, 3632741, 19843321, 109294301, 606937681,
{7, 43, 265, 1639, 10177, 63463, 397585, 2503159,15842497, 100819783, 645272305,
{8, 57, 407, 2913, 20903, 150417, 1085687, 7861953, 57130823, 416692977, 3051068567,
{9, 73, 593, 4825, 39329, 321193, 2628593, 21560185, 177264449, 1461162313, 12076718993,

ysr

求通项公式
{2, 03, 005, 009, 017, 033, 065, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537,

公式为：1+2^(n-1)，其中n大于等于1

Treenewbee

{3, 07, 017, 043, 113, 307, 857, 2443, 7073, 20707, 61097, 181243, 539633, 1610707, 4815737...
\[a_n=3^{n - 1} + 2^n\]

Treenewbee

{4, 13, 043, 145, 499, 1753, 6283, 22945, 85219, 321193, 1225723, 4725745, 18371539,...

\[a_n=4^{n - 1} + 3^n\]

Treenewbee

1. Table[m^(n - 1) + (m - 1)^n, {m, 2, 10}, {n, 1, 10}]

复制代码

{{2,3,5,9,17,33,65,129,257,513},
{3,7,17,43,113,307,857,2443,7073,20707},
{4,13,43,145,499,1753,6283,22945,85219,321193},{5,21,89,381,1649,7221,32009,143661,652769,3001701},{6,31,161,841,4421,23401,124781,670561,3632741,19843321},{7,43,265,1639,10177,63463,397585,2503159,15842497,100819783},{8,57,407,2913,20903,150417,1085687,7861953,57130823,416692977},{9,73,593,4825,39329,321193,2628593,21560185,177264449,1461162313},{10,91,829,7561,69049,631441,5782969,53046721,487420489,4486784401}}

王守恩

这样也可以:\(m^{n+1}+(m+1)^{n}\)

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,