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Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
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%I #15 Sep 08 2022 08:45:58

%S 0,1,2,10,29,70,148,289,534,950,1645,2794,4680,7761,12778,20930,34157,

%T 55598,90332,146577,237630,385006,623517,1009490,1634064,2644705,

%U 4280018,6926074,11207549,18135190,29344420,47481409,76827750,124311206

%N Coefficient of x in 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) - 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 G. C. Greubel, <a href="/A192976/b192976.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).

%F a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).

%F 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 a(n) = Fibonacci(n+4) + 3*Lucas(n+3) - (2*n^2 + 8*n + 15). - _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 -1;

%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}] (* A192975 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192976 *)

%t (* Additional programs *)

%t Table[Fibonacci[n+4]+3*LucasL[n+3] -(2*n^2+8*n+15), {n,0,40}] (* _G. C. Greubel_, Jul 24 2019 *)

%o (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 (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 (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 (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 Cf. A000032, A000045, A192232, A192744, A192951, A192975.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jul 13 2011