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A192774 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.

%I A192774 #14 Feb 18 2025 12:19:24
%S A192774 0,0,1,1,6,10,34,74,206,499,1301,3264,8348,21152,53828,136720,347533,
%T A192774 883157,2244462,5704094,14496130,36840606,93625542,237939591,
%U A192774 604694601,1536764208,3905506648,9925401280,25224262440,64104575344
%N A192774 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.
%C A192774 For discussions of polynomial reduction, see A192232 and A192744.
%H A192774 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-1,-5,1,1).
%F A192774 a(n) = a(n-1)+5*a(n-2)-a(n-3)-5*a(n-4)+a(n-5)+a(n-6).
%F A192774 G.f.: -x^3/(x^6+x^5-5*x^4-x^3+5*x^2+x-1). [_Colin Barker_, Nov 23 2012]
%e A192774 The first five polynomials p(n,x) and their reductions are as follows:
%e A192774 F1(x)=1 -> 1
%e A192774 F2(x)=x -> x
%e A192774 F3(x)=x^2+1 -> x^2+1
%e A192774 F4(x)=x^3+2x -> x^2+4x+1
%e A192774 F5(x)=x^4+3x^2+1 -> 6x^2+3x+2, so that
%e A192774 A192772=(1,0,1,1,2,...), A192773=(0,1,0,4,3,...), A192774=(0,0,1,1,6,...)
%t A192774 (See A192772.)
%t A192774 LinearRecurrence[{1,5,-1,-5,1,1},{0,0,1,1,6,10},30] (* _Harvey P. Dale_, Jun 25 2017 *)
%Y A192774 Cf. A192232, A192744, A192772.
%K A192774 nonn,easy
%O A192774 1,5
%A A192774 _Clark Kimberling_, Jul 09 2011