This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A095128 #19 Sep 08 2022 08:45:13
%S A095128 1,4,13,46,160,559,1951,6811,23776,82999,289738,1011436,3530785,
%T A095128 12325489,43026601,150199996,524327701,1830356494,6389524888,
%U A095128 22304959951,77863573135,271811114419,948855529576,3312325244431,11562875678026,40364421993364,140906692091713
%N A095128 a(n+3) = 3*a(n+2) + 2*a(n+1) - a(n).
%C A095128 A sequence generated from an inverse Bell matrix, M.
%C A095128 a(n)/a(n-1) tends to 3.4908636153..., which is a root of x^3 - 3*x^2 - 2*x + 1 and an eigenvalue of M. A095127 is generated from the reflected polynomial: x^3 - 2*x^2 - 3*x + 1 and the inverse matrix of M. Bell numbers are sums of row terms of the 3rd-order Stirling number of the second kind matrix shown on p. 171 of Aldrovandi, the matrix being [1 0 0 / 1 1 0 / 1 3 1]. Rotations, or inverses, or related polynomials generate A095125, A095126, A095127, A095128.
%D A095128 R. Aldrovandi, "Special Matrices of Mathematical Physics", World Scientific, 2001, section 13.3.1, "Inverting Bell Matrices", p. 171.
%H A095128 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-1).
%F A095128 Invert the matrix used to generate A095127, getting M = [3 2 -1 / 1 0 0 / 0 1 0]. Then M^n * [1 1 1] = [p q r] where a(n) = the center term q.
%F A095128 G.f.: (-x^2+x+1)/(x^3-2*x^2-3*x+1). - _Harvey P. Dale_, Dec 14 2012
%e A095128 a(6) = 559 = 3*a(5) + 2*a(4) - a(3) = 3*160 + 2*46 - 13.
%e A095128 a(4) = 46 since M^4 * [1 1 1] = [160 46 13] = [a(5) a(4) a(3)].
%t A095128 a[n_] := (MatrixPower[{{3, 2, -1}, {1, 0, 0}, {0, 1, 0}}, n].{{1}, {1}, {1}})[[2, 1]]; Table[ a[n], {n, 24}] (* _Robert G. Wilson v_, Jun 01 2004 *)
%t A095128 LinearRecurrence[{3,2,-1},{1,4,13},30] (* _Harvey P. Dale_, Dec 14 2012 *)
%o A095128 (Magma) I:=[1,4,13]; [n le 3 select I[n] else 3*Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jul 25 2015
%Y A095128 Cf. A095125, A095126, A095127.
%K A095128 nonn
%O A095128 1,2
%A A095128 _Gary W. Adamson_, May 29 2004
%E A095128 Edited and extended by _Robert G. Wilson v_, Jun 01 2004
%E A095128 a(25)-a(27) from _Vincenzo Librandi_, Jul 25 2015