A095126 Expansion of x*(4+5*x-x^2)/ (1-2*x-3*x^2+x^3).

4, 13, 37, 109, 316, 922, 2683, 7816, 22759, 66283, 193027, 562144, 1637086, 4767577, 13884268, 40434181, 117753589, 342925453, 998677492, 2908377754, 8469862531, 24666180832, 71833571503, 209195822971, 609226179619

Offset: 1

Author

Gary W. Adamson, May 29 2004

Keywords

• nonn
• easy

Comments

A sequence generated from a rotated Stirling number of the second kind matrix, companion to A095125.

a(n)/a(n-1) tends to 2.9122291784...an eigenvalue of M and a root of the characteristic polynomial x^3 - 2x^2 - 3x + 1. A095127 is generated from the same polynomial, with the reversal x^3 - 3x^2 - 2x + 1 being the characteristic polynomial of A095128.

Examples

			a(6) = 922 = 2*316 + 3*109 - 37 = 2*a(5) + 3*a(4) - a(3).
a(5) = 316 since M^5 * [1 1 1] = [202 316 69] = [A095125(6), a(n), A095125(5)]
		

References

• R. Aldrovandi, "Special Matrices of Mathematical Physics", World Scientific, 2001, Section 13.3.1, "Inverting Bell Matrices", p. 171.

Links

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

Crossrefs

Cf. A095125, A095127, A095128.

Programs

Mathematica

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {1, 0, 0}}, n].{{1}, {1}, {1}})[[2, 1]]; Table[ a[n], {n, 26}] (* Robert G. Wilson v, Jun 01 2004 *)
LinearRecurrence[{2,3,-1},{4,13,37},30] (* Harvey P. Dale, Jan 18 2016 *)

PARI

Vec((4+5*x-x^2)/(1-2*x-3*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Formula

a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n); with a(1) = 4, a(2) = 13, a(3) = 37.

Let M = a rotated Stirling number of the second kind matrix [1 1 1 / 3 1 0 / 1 0 0] (a rotation of [1 0 0 / 1 1 0 / 1 3 1]). Then M^n * [1 1 1] = [A095125(n+1), a(n), A095125(n)].

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 01 2004