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

%I A095125 #30 Sep 08 2022 08:45:13
%S A095125 1,3,8,24,69,202,587,1711,4981,14508,42248,123039,358314,1043497,
%T A095125 3038897,8849971,25773136,75057288,218584013,636566754,1853828259,
%U A095125 5398772767,15722463557,45787417156,133343452216,388326692343,1130896324178,3293429273169,9591220826529
%N A095125 Expansion of -x*(-1-x+x^2) / ( 1-2*x-3*x^2+x^3 ).
%C A095125 A sequence generated from a rotated Stirling number of the second kind matrix.
%C A095125 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.
%D A095125 R. Aldrovandi, "Special Matrices of Mathematical Physics," World Scientific, 2001, Section 13.3.1 "Inverting Bell Matrices", p. 171.
%H A095125 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-1).
%F A095125 a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n), with a(1) = 1, a(2) = 3, a(3) = 8.
%F A095125 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)].
%e A095125 a(5) = 69 = 2*a(4) + 3*a(3) - a(2) = 2*24 + 3*8 - 3.
%e A095125 a(5) = 69 since M^5 * [1 1 1] = [202 316 69] = [a(6) A095126(a) a(5)].
%t A095125 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 *)
%t A095125 LinearRecurrence[{2,3,-1},{1,3,8},30] (* _Harvey P. Dale_, Nov 13 2011 *)
%o A095125 (PARI) Vec((1+x-x^2)/(1-2*x-3*x^2+x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 25 2012
%o A095125 (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
%Y A095125 Cf. A095126, A095127, A095128.
%K A095125 nonn,easy
%O A095125 1,2
%A A095125 _Gary W. Adamson_, May 29 2004
%E A095125 Edited, corrected and extended by _Robert G. Wilson v_, Jun 01 2004
%E A095125 Definition corrected by _Harvey P. Dale_, Nov 13 2011
%E A095125 a(27)-a(29) from _Vincenzo Librandi_, Jul 25 2015