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A192777 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1. See Comments.

%I A192777 #8 Jun 13 2015 00:53:54
%S A192777 1,0,1,1,2,8,14,55,121,392,989,2912,7797,22104,60553,169289,467622,
%T A192777 1300888,3603914,10008543,27755249,77034176,213702153,593005504,
%U A192777 1645265209,4565154816,12666317073,35144684065,97512548090,270561677224
%N A192777 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1. See Comments.
%C A192777 For discussions of polynomial reduction, see A192232 and A192744.
%H A192777 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-1,-6,1,1).
%F A192777 a(n)=a(n-1)+6*a(n-2)-a(n-3)-6*a(n-4)+a(n-5)+a(n-6).
%F A192777 G.f.: -x*(1-5*x^2+x^4-x+x^3) / ( (x^2-x-1)*(x^4+2*x^3-3*x^2-2*x+1) ). - _R. J. Mathar_, May 06 2014
%e A192777 The first five polynomials p(n,x) and their reductions are as follows:
%e A192777 F1(x)=1 -> 1
%e A192777 F2(x)=x -> x
%e A192777 F3(x)=x^2+1 -> x^2+1
%e A192777 F4(x)=x^3+2x -> x^2+5x+1
%e A192777 F5(x)=x^4+3x^2+1 -> 7x^2+4x+2, so that
%e A192777 A192777=(1,0,1,1,2,...), A192778=(0,1,0,5,4,...), A192779=(0,0,1,1,7,...)
%t A192777 q = x^3; s = x^2 + 3 x + 1; z = 40;
%t A192777 p[n_, x_] := Fibonacci[n, x];
%t A192777 Table[Expand[p[n, x]], {n, 1, 7}]
%t A192777 reduce[{p1_, q_, s_, x_}] :=
%t A192777 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192777        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192777 t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];
%t A192777 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192777   (* A192777 *)
%t A192777 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192777   (* A192778 *)
%t A192777 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
%t A192777   (* A192779 *)
%Y A192777 Cf. A192744, A192232, A192616, A192772, A192778, A192779.
%K A192777 nonn
%O A192777 1,5
%A A192777 _Clark Kimberling_, Jul 09 2011