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A192975 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192975 #15 Sep 08 2022 08:45:58
%S A192975 1,1,8,19,41,78,141,245,416,695,1149,1886,3081,5017,8152,13227,21441,
%T A192975 34734,56245,91053,147376,238511,385973,624574,1010641,1635313,
%U A192975 2646056,4281475,6927641,11209230,18136989,29346341,47483456,76829927
%N A192975 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192975 The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) - 1 + 2*n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.
%H A192975 G. C. Greubel, <a href="/A192975/b192975.txt">Table of n, a(n) for n = 0..1000</a>
%H A192975 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192975 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F A192975 G.f.: (1-2*x+7*x^2-2*x^3)/((1-x-x^2)*(1-x)^2). - _R. J. Mathar_, May 11 2014
%F A192975 a(n) = Fibonacci(n+3) + 3*Lucas(n+2) - 2*(2*n+5). - _G. C. Greubel_, Jul 24 2019
%t A192975 (* First program *)
%t A192975 q = x^2; s = x + 1; z = 40;
%t A192975 p[0, x]:= 1;
%t A192975 p[n_, x_]:= x*p[n-1, x] +2*n^2 -1;
%t A192975 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192975 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192975 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192975 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192975 *)
%t A192975 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192976 *)
%t A192975 (* Additional programs *)
%t A192975 Table[Fibonacci[n+3]+3*LucasL[n+2] -2*(2*n+5), {n,0,40}] (* _G. C. Greubel_, Jul 24 2019 *)
%o A192975 (PARI) vector(40, n, n--; f=fibonacci; 4*f(n+3)+3*f(n+1) -2*(2*n+5)) \\ _G. C. Greubel_, Jul 24 2019
%o A192975 (Magma) [Fibonacci(n+3)+3*Lucas(n+2)-2*(2*n+5): n in [0..40]]; // _G. C. Greubel_, Jul 24 2019
%o A192975 (Sage) f=fibonacci; [4*f(n+3)+3*f(n+1) -2*(2*n+5) for n in (0..40)] # _G. C. Greubel_, Jul 24 2019
%o A192975 (GAP) F:=Fibonacci;; List([0..40], n-> 4*F(n+3)+3*F(n+1) -2*(2*n+5)); # _G. C. Greubel_, Jul 24 2019
%Y A192975 Cf. A000032, A000045, A192232, A192744, A192951, A192976.
%K A192975 nonn
%O A192975 0,3
%A A192975 _Clark Kimberling_, Jul 13 2011