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Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1.
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%I #13 Feb 23 2021 14:13:44

%S 0,1,0,3,2,10,16,43,92,213,486,1100,2522,5719,13068,29721,67772,

%T 154334,351670,801137,1825184,4158219,9473244,21582392,49169220,

%U 112018989,255203904,581412535,1324587918,3017709810,6875021540,15662845615

%N Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+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 (1,4,-1,-4,1,1).

%F a(n) = a(n-1)+4*a(n-2)-a(n-3)-4a(n-4)+a(n-5)+a(n-6).

%F G.f.: x^2*(x^2+x-1)/(x^6+x^5-4*x^4-x^3+4*x^2+x-1). [_Colin Barker_, Jul 27 2012]

%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+3x+1

%e F5(x)=x^4+3x^2+1 -> 4x^2+2x+2, so that

%e A192616=(1,0,1,1,2,...), A192617=(0,1,0,3,2,...), A192651=(0,0,1,1,5,...)

%t (See A192616.)

%t LinearRecurrence[{1,4,-1,-4,1,1},{0,1,0,3,2,10},40] (* _Harvey P. Dale_, Feb 23 2021 *)

%Y Cf. A192744, A192232, A192616.

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, Jul 09 2011