A228569 -id:A228569 - cp's OEIS frontend

A228840 a(n) = 3^n*A228569(n).

2, 15, 171, 2160, 27783, 358425, 4626234, 59716035, 770832207, 9950150160, 128439782811, 1657942687845, 21401266181778, 276254540154855, 3565983915414819, 46030886147041200, 594181726489417887, 7669891971371155905, 99005472955353055626

Offset: 0

R. J. Mathar, Nov 10 2013

Bhadouria et al. call this the 3-binomial transform of the 3-Lucas sequence.

P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence T_3.
Index entries for linear recurrences with constant coefficients, signature (15,-27).

LinearRecurrence[{15,-27},{2,15},30] (* Harvey P. Dale, May 20 2018 *)

G.f.: ( 2-15*x ) / ( 1-15*x+27*x^2 ).

Binomial transform of the binomial transform of A057076.

A309853 Array read by antidiagonals: ((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k for columns k >= 0 and rows n >= 0, where x = 4*n+1 and y = ceiling(sqrt(x)) and z = y+1-(y mod 2).

Original entry on oeis.org

2, 1, 2, 1, 3, 2, 1, 7, 3, 2, 1, 18, 9, 5, 2, 1, 47, 27, 19, 5, 2, 1, 123, 81, 80, 21, 5, 2, 1, 322, 243, 343, 95, 23, 5, 2, 1, 843, 729, 1475, 433, 110, 25, 7, 2, 1, 2207, 2187, 6346, 1975, 527, 125, 39, 7, 2, 1, 5778, 6561, 27305, 9009, 2525, 625, 238, 41, 7, 2

Offset: 0

Charles L. Hohn, Aug 20 2019

One of 4 related arrays (the others being A191347, A191348, and A309852) where the two halves of the main formula approach the integers shown and 0 respectively, and also with A309852 where rows represent various Fibonacci series a(n) = a(n-2)*b + a(n-1)*c where b and c are integers >= 0.

			2, 1,  1,   1,    1,     1,     1,      1,       1,        1, ...
2, 3,  7,  18,   47,   123,   322,    843,    2207,     5778, ...
2, 3,  9,  27,   81,   243,   729,   2187,    6561,    19683, ...
2, 5, 19,  80,  343,  1475,  6346,  27305,  117487,   505520, ...
2, 5, 21,  95,  433,  1975,  9009,  41095,  187457,   855095, ...
2, 5, 23, 110,  527,  2525, 12098,  57965,  277727,  1330670, ...
2, 5, 25, 125,  625,  3125, 15625,  78125,  390625,  1953125, ...
2, 7, 39, 238, 1471,  9107, 56394, 349223, 2162591, 13392022, ...
2, 7, 41, 259, 1649, 10507, 66953, 426643, 2718689, 17324251, ...
2, 7, 43, 280, 1831, 11977, 78346, 512491, 3352399, 21929320, ...
2, 7, 45, 301, 2017, 13517, 90585, 607061, 4068257, 27263677, ...
...

Row 2 is A005248, row 3 (except the first term) is A000244, row 4 is A228569, row 5 is A159289, row 6 is A003501, row 7 (except the first term) is A000351.

Cf. A191347, A191348, and A309852.

T(n, k) = my(x = 4*n+1, y = ceil(sqrt(x)), z = y+1-(y % 2)); round(((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k);
matrix(9, 9, n, k, T(n-1, k-1)) \\ Michel Marcus, Aug 22 2019

Revised orientation of n and k to customary T(n, k), by Charles L. Hohn, Sep 27 2024

cp's OEIS Frontend

A228840 a(n) = 3^n*A228569(n).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A309853 Array read by antidiagonals: ((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k for columns k >= 0 and rows n >= 0, where x = 4*n+1 and y = ceiling(sqrt(x)) and z = y+1-(y mod 2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions