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Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1. See Comments.
14

%I #14 Mar 03 2022 13:07:32

%S 1,0,1,1,2,6,10,29,57,142,309,720,1625,3714,8457,19259,43902,99970,

%T 227830,518943,1182401,2693624,6136837,13980960,31851853,72565704,

%U 165320833,376638417,858066430,1954869262,4453630790,10146374277,23115721705

%N Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1. See Comments.

%C For discussions of polynomial reduction, see A192232 and A192744.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-1,-4,1,1).

%F a(n) = a(n-1)+4*a(n-2)-a(n-3)-4a(n-4)+a(n-5)+a(n-6).

%F G.f.: -x*(x^4+x^3-3*x^2-x+1)/(x^6+x^5-4*x^4-x^3+4*x^2+x-1). [_Colin Barker_, Jul 27 2012]

%e The first five polynomials p(n,x) and their reductions:

%e F1(x)=1 -> 1

%e F2(x)=x -> x

%e F3(x)=x^2+1 -> x^2+1

%e F4(x)=x^3+2x -> x^2+3x+1

%e F5(x)=x^4+3x^2+1 -> 4x^2+2x+2, so that

%e A192616=(1,0,1,1,2,...), A192617=(0,1,0,3,2,...), A192651=(0,0,1,1,5,...)

%t q = x^3; s = x^2 + x + 1; z = 40;

%t p[n_, x_] := Fibonacci[n, x];

%t Table[Expand[p[n, x]], {n, 1, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 +

%t PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

%t (* A192616 *)

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

%t (* A192617 *)

%t u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]

%t (* A192651 *)

%Y Cf. A192744, A192232, A192617, A192651.

%K nonn,easy

%O 1,5

%A _Clark Kimberling_, Jul 09 2011