A095125 Expansion of -x*(-1-x+x^2) / ( 1-2*x-3*x^2+x^3 ).

1, 3, 8, 24, 69, 202, 587, 1711, 4981, 14508, 42248, 123039, 358314, 1043497, 3038897, 8849971, 25773136, 75057288, 218584013, 636566754, 1853828259, 5398772767, 15722463557, 45787417156, 133343452216, 388326692343, 1130896324178, 3293429273169, 9591220826529

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.

a(n)/a(n-1) tends to the largest positive eigenvalue of the matrix, 2.9122291784..., a root of the characteristic polynomial x^3 - 2x^2 - 3x + 1; e.g., a(9)/a(8) = 4981/1711 = 2.91116... A095127 is generated from an inverse of M, while A095126 is generated from M.

Examples

			a(5) = 69 = 2*a(4) + 3*a(3) - a(2) = 2*24 + 3*8 - 3.
a(5) = 69 since M^5 * [1 1 1] = [202 316 69] = [a(6) A095126(a) a(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. A095126, A095127, A095128.

Programs

Magma

I:=[1,3,8]; [n le 3 select I[n]  else 2*Self(n-1)+3*Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 25 2015

Mathematica

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {1, 0, 0}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 25}] (* Robert G. Wilson v, Jun 01 2004 *)
LinearRecurrence[{2,3,-1},{1,3,8},30] (* Harvey P. Dale, Nov 13 2011 *)

PARI

Vec((1+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) = 1, a(2) = 3, a(3) = 8.

M = [1 1 1 / 3 1 0 / 1 0 0], a rotation of a Stirling number of the second kind matrix [1 0 0 / 1 1 0 / 1 3 1]; then M^n * [1 1 1] = [a(n+1), A095126(n) a(n)].

Extensions

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

Definition corrected by Harvey P. Dale, Nov 13 2011

a(27)-a(29) from Vincenzo Librandi, Jul 25 2015