cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A155559 a(n) = 2*A131577(n).

Original entry on oeis.org

0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592
Offset: 0

Views

Author

Paul Curtz, Jan 24 2009

Keywords

Comments

Essentially the same as A131577, A046055, A011782, A000079 and A034008.

Links

Crossrefs

Cf. A000079, A007582, A011782, A034008, A046055, A063376, A084633, A090129, A131577, A137173.

Programs

Formula

a(n) = A000079(n), n>0.
a(n) = (-1)^(n+1)*A084633(n+1).
a(n) + A155543(n) = 2^n+4^n = A063376(n) = 2*A007582(n) =2*A137173(2n+1).
Conjecture: a(n) = A090129(n+3)-A090129(n+2).
G.f.: 2*x/(1-2*x). - R. J. Mathar, Jul 23 2009
E.g.f.: exp(2*x) - 1. - Stefano Spezia, Aug 26 2025

Extensions

Edited by R. J. Mathar, Jul 23 2009
Extended by Omar E. Pol, Nov 19 2012

A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1
Offset: 0

Views

Author

Charles L. Hohn, May 31 2011

Keywords

Examples

			1, 0,  0,   0,    0,    0,     0,      0,       0,        0,        0, ...
1, 1,  2,   4,    8,   16,    32,     64,     128,      256,      512, ...
1, 1,  3,   7,   17,   41,    99,    239,     577,     1393,     3363, ...
1, 1,  4,  10,   28,   76,   208,    568,    1552,     4240,    11584, ...
1, 2,  8,  32,  128,  512,  2048,   8192,   32768,   131072,   524288, ...
1, 2,  9,  38,  161,  682,  2889,  12238,   51841,   219602,   930249, ...
1, 2, 10,  44,  196,  872,  3880,  17264,   76816,   341792,  1520800, ...
1, 2, 11,  50,  233, 1082,  5027,  23354,  108497,   504050,  2341691, ...
1, 2, 12,  56,  272, 1312,  6336,  30592,  147712,   713216,  3443712, ...
1, 3, 18, 108,  648, 3888, 23328, 139968,  839808,  5038848, 30233088, ...
1, 3, 19, 117,  721, 4443, 27379, 168717, 1039681,  6406803, 39480499, ...
1, 3, 20, 126,  796, 5028, 31760, 200616, 1267216,  8004528, 50561600, ...
1, 3, 21, 135,  873, 5643, 36477, 235791, 1524177,  9852435, 63687141, ...
1, 3, 22, 144,  952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
...

Crossrefs

Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.
Cf. A191348 which uses ceiling() in place of floor().

Programs

  • PARI
    T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
matrix(9,9, n, k, T(n-1,k-1)) \\ Michel Marcus, Aug 22 2019
  • PARI
    T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 22 2019

Formula

For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019
T(n, k) = Sum_{i=0..floor((k+1)/2)} binomial(k, 2*i)*floor(sqrt(n))^(k-2*i)*n^i for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019

A191348 Array read by antidiagonals: ((ceiling(sqrt(n)) + sqrt(n))^k + (ceiling(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 2, 1, 0, 8, 20, 7, 2, 1, 0, 16, 68, 26, 8, 3, 1, 0, 32, 232, 97, 32, 14, 3, 1, 0, 64, 792, 362, 128, 72, 15, 3, 1, 0, 128, 2704, 1351, 512, 376, 81, 16, 3, 1, 0
Offset: 0

Views

Author

Charles L. Hohn, May 31 2011

Keywords

Examples

			1, 0,  0,   0,    0,     0,      0,      0,       0,        0,         0, ...
1, 1,  2,   4,    8,    16,     32,     64,     128,      256,       512, ...
1, 2,  6,  20,   68,   232,    792,   2704,    9232,    31520,    107616, ...
1, 2,  7,  26,   97,   362,   1351,   5042,   18817,    70226,    262087, ...
1, 2,  8,  32,  128,   512,   2048,   8192,   32768,   131072,    524288, ...
1, 3, 14,  72,  376,  1968,  10304,  53952,  282496,  1479168,   7745024, ...
1, 3, 15,  81,  441,  2403,  13095,  71361,  388881,  2119203,  11548575, ...
1, 3, 16,  90,  508,  2868,  16192,  91416,  516112,  2913840,  16450816, ...
1, 3, 17,  99,  577,  3363,  19601, 114243,  665857,  3880899,  22619537, ...
1, 3, 18, 108,  648,  3888,  23328, 139968,  839808,  5038848,  30233088, ...
1, 4, 26, 184, 1316,  9424,  67496, 483424, 3462416, 24798784, 177615776, ...
1, 4, 27, 196, 1433, 10484,  76707, 561236, 4106353, 30044644, 219825387, ...
1, 4, 28, 208, 1552, 11584,  86464, 645376, 4817152, 35955712, 268377088, ...
1, 4, 29, 220, 1673, 12724,  96773, 736012, 5597777, 42574180, 323800109, ...
1, 4, 30, 232, 1796, 13904, 107640, 833312, 6451216, 49943104, 386642400, ...
...

Crossrefs

Row 1 is A000007, row 2 is A011782, row 3 is A006012, row 4 is A001075, row 5 is A081294, row 6 is A098648, row 7 is A084120, row 8 is A146963, row 9 is A001541, row 10 is A081341, row 11 is A084134, row 13 is A090965.
Row 3*2 is A056236, row 4*2 is A003500, row 5*2 is A155543, row 9*2 is A003499.
Cf. A191347 which uses floor() in place of ceiling().

Programs

  • PARI
    T(n, k) = if (k==0, 1, if (k==1, ceil(sqrt(n)), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 23 2019

Formula

For each row n >= 0 let T(n,0)=1 and T(n,1) = ceiling(sqrt(n)), then for each column k >= 2: T(n,k) = T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 23 2019
Showing 1-3 of 3 results.

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