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

Author

Gary W. Adamson, Jun 02 2004

Keywords

• nonn
• easy

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.

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

Formula

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

Extensions

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

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