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A192874 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

%I A192874 #15 Sep 08 2022 08:45:58
%S A192874 1,0,4,6,26,72,246,774,2532,8150,26434,85448,276654,895054,2896788,
%T A192874 9373678,30334682,98163784,317666758,1027987894,3326644036,
%U A192874 10765237670,34837054674,112735054856,364818336766,1180576879422
%N A192874 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.
%C A192874 The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1.  See A192872.
%H A192874 G. C. Greubel, <a href="/A192874/b192874.txt">Table of n, a(n) for n = 0..1000</a>
%H A192874 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,6,-5,-6,4).
%F A192874 a(n) = 2*a(n-1) + 6*a(n-2) - 5*a(n-3) - 6*a(n-4) + 4*a(n-5).
%F A192874 G.f.: (x^2-x+1)*(4*x^2+x-1) / ( (x-1)*(x^2-x-1)*(4*x^2+2*x-1) ). - _R. J. Mathar_, May 06 2014
%t A192874 q = x^2; s = x + 1; z = 26;
%t A192874 p[0, x_] := 1; p[1, x_] := x;
%t A192874 p[n_, x_] := p[n - 1, x]*x + 2*p[n - 2, x]*x^2 + 1;
%t A192874 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192874 reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192874 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192874 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192874 *)
%t A192874 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192875 *)
%t A192874 LinearRecurrence[{2,6,-5,-6,4}, {1,0,4,6,26}, 30] (* _G. C. Greubel_, Jan 08 2019 *)
%o A192874 (PARI) my(x='x+O('x^30)); Vec((x^2-x+1)*(4*x^2+x-1)/((x-1)*(x^2-x-1)*( 4*x^2+2*x-1))) \\ _G. C. Greubel_, Jan 08 2019
%o A192874 (Magma) I:=[1,0,4,6,26]; [n le 5 select I[n] else 2*Self(n-1)+6*Self(n-2) -5*Self(n-3)-6*Self(n-4)+4*Self(n-5): n in [1..30]]; // _G. C. Greubel_, Jan 08 2019
%o A192874 (Sage) ((x^2-x+1)*(4*x^2+x-1)/((x-1)*(x^2-x-1)*( 4*x^2+2*x-1))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 08 2019
%o A192874 (GAP) a:=[1,0,4,6,26];; 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 A192874 Cf. A192872, A192875.
%K A192874 nonn
%O A192874 0,3
%A A192874 _Clark Kimberling_, Jul 11 2011