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A192782 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+1.

%I A192782 #12 Feb 18 2025 12:37:43
%S A192782 0,0,1,1,4,6,14,26,52,103,201,400,784,1552,3056,6032,11897,23465,
%T A192782 46292,91302,180110,355258,700772,1382287,2726609,5378336,10608928,
%U A192782 20926496,41278176,81422624,160608817,316806289,624911012,1232657862,2431458958
%N A192782 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+1.
%C A192782 For discussions of polynomial reduction, see A192232 and A192744.
%H A192782 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,-3,1,1).
%F A192782 a(n) = a(n-1)+3*a(n-2)-a(n-3)-3*a(n-4)+a(n-5)+a(n-6).
%F A192782 G.f.: -x^3/(x^6+x^5-3*x^4-x^3+3*x^2+x-1). [_Colin Barker_, Nov 23 2012]
%e A192782 The first five polynomials p(n,x) and their reductions:
%e A192782 F1(x)=1 -> 1
%e A192782 F2(x)=x -> x
%e A192782 F3(x)=x^2+1 -> x^2+1
%e A192782 F4(x)=x^3+2x -> x^2+2x+1
%e A192782 F5(x)=x^4+3x^2+1 -> 4x^2+1x+2, so that
%e A192782 A192777=(1,0,1,1,2,...), A192778=(0,1,0,2,1,...), A192779=(0,0,1,1,4,...)
%t A192782 (See A192780.)
%Y A192782 Cf. A192744, A192232, A192616, A192780, A192781.
%K A192782 nonn,easy
%O A192782 1,5
%A A192782 _Clark Kimberling_, Jul 09 2011