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A192779 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1.

%I A192779 #14 Feb 18 2025 12:37:30
%S A192779 0,0,1,1,7,12,47,107,337,868,2520,6808,19192,52756,147185,407069,
%T A192779 1131599,3136292,8707655,24151335,67025633,185946904,515971328,
%U A192779 1431563056,3972149312,11021051864,30579529249,84846231017,235416993159,653192251196
%N A192779 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1.
%C A192779 For discussions of polynomial reduction, see A192232 and A192744.
%H A192779 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-1,-6,1,1).
%F A192779 a(n) = a(n-1)+6*a(n-2)-a(n-3)-6*a(n-4)+a(n-5)+a(n-6).
%F A192779 G.f.: -x^3/((x^2-x-1)*(x^4+2*x^3-3*x^2-2*x+1)). [_Colin Barker_, Nov 23 2012]
%e A192779 The first five polynomials p(n,x) and their reductions:
%e A192779 F1(x)=1 -> 1
%e A192779 F2(x)=x -> x
%e A192779 F3(x)=x^2+1 -> x^2+1
%e A192779 F4(x)=x^3+2x -> x^2+5x+1
%e A192779 F5(x)=x^4+3x^2+1 -> 7x^2+4x+2, so that
%e A192779 A192777=(1,0,1,1,2,...), A192778=(0,1,0,5,4,...), A192779=(0,0,1,1,7,...)
%t A192779 (See A192777.)
%t A192779 LinearRecurrence[{1,6,-1,-6,1,1},{0,0,1,1,7,12}, 30] (* _Harvey P. Dale_, Oct 29 2018 *)
%Y A192779 Cf. A192744, A192232, A192616, A192772, A192777.
%K A192779 nonn,easy
%O A192779 1,5
%A A192779 _Clark Kimberling_, Jul 09 2011