Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Feb 08 2022 13:39:15
%S 1,4,10,31,88,259,751,2191,6376,18574,54085,157516,458713,1335889,
%T 3890401,11329756,32994826,96088519,279831760,814934251,2373275263,
%U 6911521519,20127934576,58617158446,170706599101,497136738964
%N a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n); with a(1) = 1, a(2) = 4, a(3) = 10.
%C A sequence generated from the characteristic polynomial of A095125 and A095126.
%C a(n)/a(n-1) tends to a 2.9122291784..., a root of the polynomial x^3 - 2x^2 - 3x + 1; e.g. a(16)/a(15) = 11329756/3890401 = 2.912233...
%H Colin Barker, <a href="/A095127/b095127.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-1).
%F M = a matrix having the same eigenvalues as the roots of the characteristic polynomial of A095125 and A095126: (x^3 - 2x^2 - 3x + 1). Then M^n * [1 1 1] = [p q r] where q = a(n) and p, r, are offset members of the same sequence.
%F G.f.: x*(1 + 2*x - x^2) / (1 - 2*x - 3*x^2 + x^3). - _Colin Barker_, Aug 31 2019
%e a(7) = 751 = 2*a(6) + 3*a(5) - a(4) = 2*259 + 3*88 - 31.
%e a(4) = 31 = center term in M^4 * [1 1 1] = [10 31 88].
%t a[1] = 1; a[2] = 4; a[3] = 10; a[n_] := a[n] = 2a[n - 1] + 3a[n - 2] - a[n - 3]; Table[ a[n], {n, 25}] (* _Robert G. Wilson v_, Jun 01 2004 *)
%t nxt[{a_,b_,c_}]:={b,c,2c+3b-a}; NestList[nxt,{1,4,10},30][[All,1]] (* or *) LinearRecurrence[{2,3,-1},{1,4,10},30] (* _Harvey P. Dale_, Feb 08 2022 *)
%o (PARI) Vec(x*(1 + 2*x - x^2) / (1 - 2*x - 3*x^2 + x^3) + O(x^30)) \\ _Colin Barker_, Aug 31 2019
%Y Cf. A095125, A095126, A095128.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, May 29 2004
%E Edited, corrected and extended by _Robert G. Wilson v_, Jun 01 2004