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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 #18 Sep 08 2022 08:45:58

%S 0,1,2,6,15,34,70,135,248,440,761,1292,2164,3589,5910,9682,15803,

%T 25726,41802,67835,109980,178196,288597,467256,756360,1224169,1981130,

%U 3205950,5187783,8394490,13583086,21978447,35562464,57541904,93105425

%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 - n + 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 Vincenzo Librandi, <a href="/A192980/b192980.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+3*x^2)/((1-x-x^2)*(1-x)^3). - _R. J. Mathar_, May 12 2014

%F a(n) = Fibonacci(n+4) + Lucas(n+3) - (n^2 + 3*n + 7). - _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] +n^2-n+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}] (* A192979 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192980 *)

%t (* Additional programs *)

%t CoefficientList[Series[x*(1-2*x+3*x^2)/((1-x-x^2)*(1-x)^3), {x,0,40}], x] (* _Vincenzo Librandi_, May 13 2014 *)

%t Table[Fibonacci[n+4]+LucasL[n+3] -(n^2+3*n+7), {n,0,40}] (* _G. C. Greubel_, Jul 24 2019 *)

%o (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+4)+f(n+2) -(n^2+3*n+7)) \\ _G. C. Greubel_, Jul 24 2019

%o (Magma) [Fibonacci(n+4)+Lucas(n+3)-(n^2+3*n+7): n in [0..40]]; // _G. C. Greubel_, Jul 24 2019

%o (Sage) f=fibonacci; [2*f(n+4)+f(n+2) -(n^2+3*n+7) for n in (0..40)] # _G. C. Greubel_, Jul 24 2019

%o (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+4)+F(n+2) -(n^2+3*n+7)); # _G. C. Greubel_, Jul 24 2019

%Y Cf. A000032, A000045, A192232, A192744, A192951, A192979.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jul 13 2011