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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 #16 Sep 08 2022 08:45:58

%S 0,1,4,11,26,55,108,201,360,627,1070,1799,2992,4937,8100,13235,21562,

%T 35055,56908,92289,149560,242251,392254,634991,1027776,1663345,

%U 2691748,4355771,7048250,11404807,18453900,29859609,48314472,78175107,126490670

%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) + 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="/A192961/b192961.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) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).

%F From _R. J. Mathar_, May 09 2014: (Start)

%F G.f.: x*(1+x)*(1-x+x^2)/((1-x-x^2)*(1-x)^3).

%F a(n) - a(n-1)= A192960(n-1). (End)

%F a(n) = 2*Fibonacci(n+5) - (n^2 + 4*n + 10). - _G. C. Greubel_, Jul 12 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] + n^2 + 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}] (* A192960 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192961 *)

%t (* Second program *)

%t With[{F=Fibonacci}, Table[2*F[n+5]-(n^2+4*n+10), {n,0,40}]] (* _G. C. Greubel_, Jul 12 2019 *)

%o (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+5)-(n^2+4*n+10)) \\ _G. C. Greubel_, Jul 12 2019

%o (Magma) F:=Fibonacci; [2*F(n+5)-(n^2+4*n+10): n in [0..40]]; // _G. C. Greubel_, Jul 12 2019

%o (Sage) f=fibonacci; [2*f(n+5)-(n^2+4*n+10) for n in (0..40)] # _G. C. Greubel_, Jul 12 2019

%o (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+5)-(n^2+4*n+10)); # _G. C. Greubel_, Jul 12 2019

%Y Cf. A000045, A192232, A192744, A192951, A192960.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jul 13 2011