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A192972 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192972 #17 Sep 08 2022 08:45:58
%S A192972 0,1,3,12,33,77,160,309,567,1004,1733,2937,4912,8137,13387,21916,
%T A192972 35753,58181,94512,153341,248575,402716,652173,1055857,1709088,
%U A192972 2766097,4476435,7243884,11721777,18967229,30690688,49659717,80352327,130014092
%N A192972 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192972 The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 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 A192972 G. C. Greubel, <a href="/A192972/b192972.txt">Table of n, a(n) for n = 0..1000</a>
%H A192972 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A192972 a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
%F A192972 G.f.: x*(1-x+5*x^2-x^3)/((1-x-x^2)*(1-x)^3). - _R. J. Mathar_, May 11 2014
%F A192972 a(n) = 4*Fibonacci(n+4) + Lucas(n+3) - 2*(n^2+4*n+8). - _G. C. Greubel_, Jul 24 2019
%t A192972 (* First program *)
%t A192972 q = x^2; s = x + 1; z = 40;
%t A192972 p[0, x]:= 1;
%t A192972 p[n_, x_]:= x*p[n-1, x] + 2*n^2;
%t A192972 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192972 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192972 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192972 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192971 *)
%t A192972 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192972 *)
%t A192972 (* Additional programs *)
%t A192972 With[{F = Fibonacci}, Table[5*F[n+4]+F[n+2] -2*(n^2+4*n+8), {n,0,40}]] (* _G. C. Greubel_, Jul 24 2019 *)
%o A192972 (PARI) vector(40, n, n--; f=fibonacci; 5*f(n+4)+f(n+2) -2*(n^2+4*n+8)) \\ _G. C. Greubel_, Jul 24 2019
%o A192972 (Magma) F:=Fibonacci; [5*F(n+4)+F(n+2) -2*(n^2+4*n+8): n in [0..40]]; // _G. C. Greubel_, Jul 24 2019
%o A192972 (Sage) f=fibonacci; [5*f(n+4)+f(n+2) -2*(n^2+4*n+8) for n in (0..40)] # _G. C. Greubel_, Jul 24 2019
%o A192972 (GAP) F:=Fibonacci;; List([0..40], n-> 5*F(n+4)+F(n+2) -2*(n^2+4*n+8)); # _G. C. Greubel_, Jul 24 2019
%Y A192972 Cf. A000032, A000045, A192232, A192744, A192951, A192971.
%K A192972 nonn
%O A192972 0,3
%A A192972 _Clark Kimberling_, Jul 13 2011