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A095125
Expansion of -x*(-1-x+x^2) / ( 1-2*x-3*x^2+x^3 ).
4
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.
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
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