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%I #17 Sep 08 2022 08:45:58
%S 1,2,9,21,44,83,149,258,437,729,1204,1975,3225,5250,8529,13837,22428,
%T 36331,58829,95234,154141,249457,403684,653231,1057009,1710338,
%U 2767449,4477893,7245452,11723459,18969029,30692610,49661765,80354505
%N Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C 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 G. C. Greubel, <a href="/A192971/b192971.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F G.f.: (1-x+5*x^2-x^3)/((1-x-x^2)*(1-x)^2). - _R. J. Mathar_, May 11 2014
%F a(n) = 4*Fibonacci(n+3) + Lucas(n+2) - 2*(2*n+5). - _G. C. Greubel_, Jul 24 2019
%t (* First program *)
%t q = x^2; s = x + 1; z = 40;
%t p[0, x]:= 1;
%t p[n_, x_]:= x*p[n-1, x] + 2*n^2;
%t Table[Expand[p[n, x]], {n, 0, 7}]
%t reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192971 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192972 *)
%t (* Additional programs *)
%t With[{F = Fibonacci}, Table[5*F[n+3]+F[n+1] -2*(2*n+5), {n,0,40}]] (* _G. C. Greubel_, Jul 24 2019 *)
%o (PARI) vector(40, n, n--; f=fibonacci; 5*f(n+3)+f(n+1) -2*(2*n+5)) \\ _G. C. Greubel_, Jul 24 2019
%o (Magma) F:=Fibonacci; [5*F(n+3)+F(n+1) -2*(2*n+5): n in [0..40]]; // _G. C. Greubel_, Jul 24 2019
%o (Sage) f=fibonacci; [5*f(n+3)+f(n+1) -2*(2*n+5) for n in (0..40)] # _G. C. Greubel_, Jul 24 2019
%o (GAP) F:=Fibonacci;; List([0..40], n-> 5*F(n+3)+F(n+1) -2*(2*n+5)); # _G. C. Greubel_, Jul 24 2019
%Y Cf. A000032, A000045, A192232, A192744, A192951, A192972.
%K nonn
%O 0,2
%A _Clark Kimberling_, Jul 13 2011
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