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A192876 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

%I A192876 #22 Sep 08 2022 08:45:58
%S A192876 1,1,4,9,31,94,309,989,3212,10373,33595,108670,351729,1138113,3683172,
%T A192876 11918737,38570247,124815294,403911805,1307084405,4229816636,
%U A192876 13687969901,44295207939,143342292894,463865421721,1501100008249
%N A192876 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.
%C A192876 The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x + 1, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1.  See A192872.
%H A192876 G. C. Greubel, <a href="/A192876/b192876.txt">Table of n, a(n) for n = 0..1000</a>
%H A192876 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,6,-5,-6,4).
%F A192876 a(n) = 2*a(n-1) + 6*a(n-2) - 5*a(n-3) - 6*a(n-4) + 4*a(n-5).
%F A192876 G.f.: (1-x-4*x^2) / ( (1-x)*(1+x-x^2)*(1-2*x-4*x^2) ). - _R. J. Mathar_, May 06 2014
%p A192876 seq(coeff(series((1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Jan 08 2019
%t A192876 q = x^2; s = x + 1; z = 26;
%t A192876 p[0, x_] := 1; p[1, x_] := x + 1;
%t A192876 p[n_, x_] := p[n - 1, x]*x + 2*p[n - 2, x]*x^2 + 1;
%t A192876 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192876 reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192876 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192876 u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]  (* A192876 *)
%t A192876 u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192877 *)
%t A192876 FindLinearRecurrence[u0]
%t A192876 FindLinearRecurrence[u1]
%t A192876 LinearRecurrence[{2,6,-5,-6,4},{1,1,4,9,31},26] (* _Ray Chandler_, Aug 02 2015 *)
%o A192876 (PARI) my(x='x+O('x^30)); Vec((1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2) )) \\ _G. C. Greubel_, Jan 08 2019
%o A192876 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)) )); // _G. C. Greubel_, Jan 08 2019
%o A192876 (Sage) ((1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2))).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 08 2019
%o A192876 (GAP) a:=[1,1,4,9,31];; for n in [6..30] do a[n]:=2*a[n-1]+6*a[n-2] - 5*a[n-3]-6*a[n-4]+4*a[n-5]; od; a; # _G. C. Greubel_, Jan 08 2019
%Y A192876 Cf. A192872, A192877.
%K A192876 nonn
%O A192876 0,3
%A A192876 _Clark Kimberling_, Jul 11 2011