%I #10 Jun 13 2015 00:53:53
%S 1,0,1,1,2,7,12,41,86,247,585,1548,3849,9896,25001,63724,161721,
%T 411257,1044878,2655719,6748972,17151849,43589578,110777391,281529169,
%U 715471992,1818293377,4620978640,11743694657,29845241080,75848270001
%N Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.
%C For discussions of polynomial reduction, see A192232 and A192744.
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-1,-5,1,1).
%F a(n)=a(n-1)+5*a(n-2)-a(n-3)-5*a(n-4)+a(n-5)+a(n-6).
%F G.f.: -x*(x^2-x-1)*(x^2+2*x-1) / (x^6+x^5-5*x^4-x^3+5*x^2+x-1). [_Colin Barker_, Jan 17 2013]
%e The first five polynomials p(n,x) and their reductions are as follows:
%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+4x+1
%e F5(x)=x^4+3x^2+1 -> 6x^2+3x+2, so that
%e A192772=(1,0,1,1,2,...), A192773=(0,1,0,4,3,...), A192774=(0,0,1,1,6,...)
%t q = x^3; s = x^2 + 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 (* A192772 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t (* A192773 *)
%t u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
%t (* A192774 *)
%Y Cf. A192744, A192232, A192616, A192773, A192774.
%K nonn,easy
%O 1,5
%A _Clark Kimberling_, Jul 09 2011