A141844 Expansion of (1+x)*(1+x^2)/((1-x)^2*(1+x+x^2)*(1-4*x)).

1, 6, 27, 113, 458, 1839, 7365, 29470, 117891, 471577, 1886322, 7545303, 30181229, 120724934, 482899755, 1931599041, 7726396186, 30905584767, 123622339093, 494489356398, 1977957425619, 7911829702505, 31647318810050, 126589275240231, 506357100960957

Offset

0,2

Comments

Old name was: a(n) = 4*a(n-1) + A042968(n), with a(0) = 1, where A042968 = "not divisible by 4": (1, 2, 3, 5, 6, 7, 9, 10, 11, ...). After the correction of a(13) the definition could be simplified. - N. J. A. Sloane, Aug 23 2018

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1660

Index entries for linear recurrences with constant coefficients, signature (5, -4, 1, -5, 4).

Formula

a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3) - 5*a(n-4) + 4*a(n-5) for n>4. - Colin Barker, Jun 26 2017

Example

a(3) = 4*a(2) + A042968(3) = 4*27 + 5 = 113.
a(13) = 4*a(12) + A042968(13) = 4*30181229 + 18 = 120724934.

Mathematica

CoefficientList[Series[(1 + x) (1 + x^2) / ((1 - x)^2 (1 + x + x^2) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 28 2017 *)
Nest[Append[#, Block[{k = #[[-1, -1]] + 1}, While[Mod[k, 4] == 0, k++]; {4 #[[-1, 1]] + k, k}]] &, {{1, 1}}, 24][[All, 1]] (* Michael De Vlieger, Jun 30 2018 *)
LinearRecurrence[{5, -4, 1, -5, 4}, {1, 6, 27, 113, 458}, 25] (* Robert G. Wilson v, Jul 28 2018 *)

Prog

(PARI) Vec((1 + x)*(1 + x^2) / ((1 - x)^2*(1 - 4*x)*(1 + x + x^2)) + O(x^30)) \\ Colin Barker, Jun 26 2017
(Magma) I:=[1, 6, 27, 113, 458]; [n le 5 select I[n] else 5*Self(n-1)-4*Self(n-2)+Self(n-3)-5*Self(n-4)+4*Self(n-5): n in [1..30]]; // Vincenzo Librandi, Jun 28 2017, Jul 28 2018

Crossrefs

Cf. A042968.

Sequence in context: A198694 A220101 A014825 * A176476 A079742 A291232

Adjacent sequences: A141841 A141842 A141843 * A141845 A141846 A141847

Keyword

nonn

Author

Gary W. Adamson, Jul 11 2008

Extensions

Corrected by Charlie Neder, Jun 22 2018

Edited by N. J. A. Sloane, Aug 23 2018, merging old entry A288571 with this one.

Status

approved