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 A095126 Expansion of x*(4+5*x-x^2)/ (1-2*x-3*x^2+x^3). 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. 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). 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)]. EXAMPLE 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)] 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 *) PROG (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 CROSSREFS Cf. A095125, A095127, A095128. Sequence in context: A080145 A097551 A224033 * A077842 A067633 A091874 Adjacent sequences:  A095123 A095124 A095125 * A095127 A095128 A095129 KEYWORD nonn,easy AUTHOR Gary W. Adamson, May 29 2004 EXTENSIONS Edited, corrected and extended by Robert G. Wilson v, Jun 01 2004 STATUS approved

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Last modified September 17 09:42 EDT 2021. Contains 347478 sequences. (Running on oeis4.)