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A192976 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192976 #15 Sep 08 2022 08:45:58
%S A192976 0,1,2,10,29,70,148,289,534,950,1645,2794,4680,7761,12778,20930,34157,
%T A192976 55598,90332,146577,237630,385006,623517,1009490,1634064,2644705,
%U A192976 4280018,6926074,11207549,18135190,29344420,47481409,76827750,124311206
%N A192976 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192976 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 A192976 G. C. Greubel, <a href="/A192976/b192976.txt">Table of n, a(n) for n = 0..1000</a>
%H A192976 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A192976 a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
%F A192976 G.f.: x*(1-2*x+7*x^2-2*x^3)/((1-x-x^2)*(1-x)^3). - _R. J. Mathar_, May 11 2014
%F A192976 a(n) = Fibonacci(n+4) + 3*Lucas(n+3) - (2*n^2 + 8*n + 15). - _G. C. Greubel_, Jul 24 2019
%t A192976 (* First program *)
%t A192976 q = x^2; s = x + 1; z = 40;
%t A192976 p[0, x]:= 1;
%t A192976 p[n_, x_]:= x*p[n-1, x] +2*n^2 -1;
%t A192976 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192976 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192976 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192976 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192975 *)
%t A192976 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192976 *)
%t A192976 (* Additional programs *)
%t A192976 Table[Fibonacci[n+4]+3*LucasL[n+3] -(2*n^2+8*n+15), {n,0,40}] (* _G. C. Greubel_, Jul 24 2019 *)
%o A192976 (PARI) vector(40, n, n--; f=fibonacci; 4*f(n+4)+3*f(n+2) -(2*n^2 + 8*n + 15)) \\ _G. C. Greubel_, Jul 24 2019
%o A192976 (Magma) [Fibonacci(n+4)+3*Lucas(n+3)-(2*n^2+8*n+15): n in [0..40]]; // _G. C. Greubel_, Jul 24 2019
%o A192976 (Sage) f=fibonacci; [4*f(n+4)+3*f(n+2) -(2*n^2+8*n+15) for n in (0..40)] # _G. C. Greubel_, Jul 24 2019
%o A192976 (GAP) F:=Fibonacci;; List([0..40], n-> 4*F(n+4)+3*F(n+2)-(2*n^2+8*n+15)); # _G. C. Greubel_, Jul 24 2019
%Y A192976 Cf. A000032, A000045, A192232, A192744, A192951, A192975.
%K A192976 nonn
%O A192976 0,3
%A A192976 _Clark Kimberling_, Jul 13 2011