A095310 a(n+3) = 2*a(n+2) + 3*(n+1) - a(n).

1, 5, 12, 38, 107, 316, 915, 2671, 7771, 22640, 65922, 191993, 559112, 1628281, 4741905, 13809541, 40216516, 117119750, 341079507, 993301748, 2892722267, 8424270271, 24533405595, 71446899736, 208069745986, 605946785585

Offset

1,2

Comments

Let M = the 3 X 3 matrix [1 1 1 / 3 1 0 / 1 0 0], then M^n * [1 0 0] = [a(n) q a(n-1] where q is another sequence with the same recursion rule.

Links

Table of n, a(n) for n=1..26.

Index entries for linear recurrences with constant coefficients, signature (2, 3, -1).

Formula

G.f.: (-x^2+3*x+1)/(x^3-3*x^2-2*x+1). - Harvey P. Dale, Jan 25 2014

Example

a(6) = 316 = 2*107 + 3*38 - 12.
a(5) = 107 since M^5 * [1 0 0] = [107 q 38].

Mathematica

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {1, 0, 0}}, n].{{1}, {0}, {0}})[[1, 1]]; Table[ a[n], {n, 27}] (* Robert G. Wilson v, Jun 05 2004 *)
LinearRecurrence[{2, 3, -1}, {1, 5, 12}, 30] (* Harvey P. Dale, Jan 25 2014 *)

Crossrefs

Cf. A001263, A006356, A006054, A061646, A061647, A095125, A095126.

Sequence in context: A297909 A220705 A034752 * A122299 A162269 A028322

Adjacent sequences: A095307 A095308 A095309 * A095311 A095312 A095313

Keyword

nonn

Author

Gary W. Adamson, Jun 02 2004

Extensions

Corrected and extended by Robert G. Wilson v, Jun 05 2004

Edited by N. J. A. Sloane, Jun 07 2004

Status

approved