A037632 Base 4 digits are, in order, the first n terms of the periodic sequence with initial period 2,1,3.

2, 9, 39, 158, 633, 2535, 10142, 40569, 162279, 649118, 2596473, 10385895, 41543582, 166174329, 664697319, 2658789278, 10635157113, 42540628455, 170162513822, 680650055289, 2722600221159, 10890400884638, 43561603538553, 174246414154215, 696985656616862, 2787942626467449, 11151770505869799

Offset

1,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,0,1,-4).

Formula

G.f.: x*(2+x+3*x^2) / ( (x-1)*(4*x-1)*(1+x+x^2) ). - R. J. Mathar, Nov 21 2011

a(n) = 4*a(n-1) + a(n-3) - 4*a(n-4). - Vincenzo Librandi, Jun 17 2012

a(n) = floor(13*4^n/21). - Christian Krause, Jun 05 2026

E.g.f.: exp(-x/2)*(-14*exp(3*x/2) + 13*exp(9*x/2) + cos(sqrt(3)*x/2) + 3*sqrt(3)*sin(sqrt(3)*x/2))/21. - Stefano Spezia, Jun 05 2026

Mathematica

LinearRecurrence[{4, 0, 1, -4}, {2, 9, 39, 158}, 40] (* Vincenzo Librandi, Jun 17 2012 *)
(* Alternative: *)
Table[FromDigits[PadRight[{}, n, {2, 1, 3}], 4], {n, 30}] (* Harvey P. Dale, Jan 01 2026 *)

Prog

(Magma) I:=[2, 9, 39, 158]; [n le 4 select I[n] else 4*Self(n-1)+Self(n-3)-4*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 17 2012
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -4, 1, 0, 4]^(n-1)*[2; 9; 39; 158])[1, 1] \\ Charles R Greathouse IV, Jun 02 2026

Crossrefs

Sequence in context: A070017 A054129 A037737 * A095939 A361358 A151009

Adjacent sequences: A037629 A037630 A037631 * A037633 A037634 A037635

Keyword

nonn,base,easy,changed

Author

Clark Kimberling

Extensions

a(24)-a(27) from Christian Krause, Jun 05 2026

Status

approved