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A192922 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

%I A192922 #16 Sep 08 2022 08:45:58
%S A192922 1,0,1,2,5,11,25,55,122,268,590,1295,2844,6240,13693,30039,65900,
%T A192922 144559,317108,695595,1525829,3346965,7341695,16104238,35325142,
%U A192922 77486710,169969295,372832346
%N A192922 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.
%C A192922 The titular polynomial is defined by p(n,x) = p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x.  For discussions of polynomial reduction, see A192232, A192744, and A192872.
%H A192922 G. C. Greubel, <a href="/A192922/b192922.txt">Table of n, a(n) for n = 0..1000</a>
%H A192922 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-3,-1).
%F A192922 a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - a(n-4).
%F A192922 G.f.: (1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4). - _R. J. Mathar_, May 08 2014
%t A192922 q = x^2; s = x + 1; z = 28;
%t A192922 p[0, x_] := 1; p[1, x_] := x;
%t A192922 p[n_, x_] := p[n - 1, x] + p[n - 2, x]*x^2;
%t A192922 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192922 reduce[{p1_, q_, s_, x_}] :=
%t A192922 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192922        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192922 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192922 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192922   (* A192922 *)
%t A192922 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192923 *)
%t A192922 LinearRecurrence[{2,2,-3,-1}, {1,0,1,2}, 30] (* _G. C. Greubel_, Feb 06 2019 *)
%o A192922 (PARI) my(x='x+O(x^30)); Vec((1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4)) \\ _G. C. Greubel_, Feb 06 2019
%o A192922 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4) )); // _G. C. Greubel_, Feb 06 2019
%o A192922 (Sage) ((1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4)).series(x, 30).coefficients(x, sparse=False)
%o A192922 (GAP) a:=[1,0,1,2];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2] -3*a[n-3]-a[n-4]; od; a; # _G. C. Greubel_, Feb 06 2019
%Y A192922 Cf. A192232, A192744, A192923.
%K A192922 nonn
%O A192922 0,4
%A A192922 _Clark Kimberling_, Jul 12 2011