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A192651 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1. See Comments.

%I A192651 #14 Feb 19 2025 02:40:59
%S A192651 0,0,1,1,5,8,23,47,113,252,578,1316,2994,6832,15545,35445,80711,
%T A192651 183928,418973,954571,2174681,4954436,11287336,25715016,58584744,
%U A192651 133468980,304072713,692745597,1578230845,3595564360,8191505015,18662090915
%N A192651 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1.  See Comments.
%C A192651 For discussions of polynomial reduction, see A192232 and A192744.
%H A192651 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-1,-4,1,1).
%F A192651 a(n) = a(n-1)+4*a(n-2)-a(n-3)-4a(n-4)+a(n-5)+a(n-6).
%F A192651 G.f.: -x^3/(x^6+x^5-4*x^4-x^3+4*x^2+x-1). [_Colin Barker_, Jul 27 2012]
%e A192651 The first five polynomials p(n,x) and their reductions are as follows:
%e A192651 F1(x)=1 -> 1
%e A192651 F2(x)=x -> x
%e A192651 F3(x)=x^2+1 -> x^2+1
%e A192651 F4(x)=x^3+2x -> x^2+3x+1
%e A192651 F5(x)=x^4+3x^2+1 -> 4x^2+2x+2, so that
%e A192651 A192616=(1,0,1,1,2,...), A192617=(0,1,0,3,2,...), A192651=(0,0,1,1,5,...)
%t A192651 (See A192616.)
%Y A192651 Cf. A192232, A192744, A192616.
%K A192651 nonn,easy
%O A192651 1,5
%A A192651 _Clark Kimberling_, Jul 09 2011