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A192616 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1. See Comments.

%I A192616 #14 Mar 03 2022 13:07:32
%S A192616 1,0,1,1,2,6,10,29,57,142,309,720,1625,3714,8457,19259,43902,99970,
%T A192616 227830,518943,1182401,2693624,6136837,13980960,31851853,72565704,
%U A192616 165320833,376638417,858066430,1954869262,4453630790,10146374277,23115721705
%N A192616 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1.  See Comments.
%C A192616 For discussions of polynomial reduction, see A192232 and A192744.
%H A192616 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-1,-4,1,1).
%F A192616 a(n) = a(n-1)+4*a(n-2)-a(n-3)-4a(n-4)+a(n-5)+a(n-6).
%F A192616 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 A192616 The first five polynomials p(n,x) and their reductions:
%e A192616 F1(x)=1 -> 1
%e A192616 F2(x)=x -> x
%e A192616 F3(x)=x^2+1 -> x^2+1
%e A192616 F4(x)=x^3+2x -> x^2+3x+1
%e A192616 F5(x)=x^4+3x^2+1 -> 4x^2+2x+2, so that
%e A192616 A192616=(1,0,1,1,2,...), A192617=(0,1,0,3,2,...), A192651=(0,0,1,1,5,...)
%t A192616 q = x^3; s = x^2 + x + 1; z = 40;
%t A192616 p[n_, x_] := Fibonacci[n, x];
%t A192616 Table[Expand[p[n, x]], {n, 1, 7}]
%t A192616 reduce[{p1_, q_, s_, x_}] :=
%t A192616 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192616        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192616 t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];
%t A192616 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192616   (* A192616 *)
%t A192616 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192616   (* A192617 *)
%t A192616 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
%t A192616   (* A192651 *)
%Y A192616 Cf. A192744, A192232, A192617, A192651.
%K A192616 nonn,easy
%O A192616 1,5
%A A192616 _Clark Kimberling_, Jul 09 2011