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

1,2

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.

References

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

Links

Table of n, a(n) for n=1..29.

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

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)].

Example

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)].

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 *)

Prog

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

Crossrefs

Cf. A095126, A095127, A095128.

Sequence in context: A091588 A297219 A018046 * A078055 A291613 A238126

Adjacent sequences: A095122 A095123 A095124 * A095126 A095127 A095128

Keyword

nonn,easy

Author

Gary W. Adamson, May 29 2004

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

Status

approved