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A095127 a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n); with a(1) = 1, a(2) = 4, a(3) = 10.

%I A095127 #14 Feb 08 2022 13:39:15
%S A095127 1,4,10,31,88,259,751,2191,6376,18574,54085,157516,458713,1335889,
%T A095127 3890401,11329756,32994826,96088519,279831760,814934251,2373275263,
%U A095127 6911521519,20127934576,58617158446,170706599101,497136738964
%N A095127 a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n); with a(1) = 1, a(2) = 4, a(3) = 10.
%C A095127 A sequence generated from the characteristic polynomial of A095125 and A095126.
%C A095127 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 A095127 Colin Barker, <a href="/A095127/b095127.txt">Table of n, a(n) for n = 1..1000</a>
%H A095127 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-1).
%F A095127 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 A095127 G.f.: x*(1 + 2*x - x^2) / (1 - 2*x - 3*x^2 + x^3). - _Colin Barker_, Aug 31 2019
%e A095127 a(7) = 751 = 2*a(6) + 3*a(5) - a(4) = 2*259 + 3*88 - 31.
%e A095127 a(4) = 31 = center term in M^4 * [1 1 1] = [10 31 88].
%t A095127 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 A095127 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 A095127 (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 A095127 Cf. A095125, A095126, A095128.
%K A095127 nonn,easy
%O A095127 1,2
%A A095127 _Gary W. Adamson_, May 29 2004
%E A095127 Edited, corrected and extended by _Robert G. Wilson v_, Jun 01 2004