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A192773 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.

%I A192773 #17 Aug 07 2025 13:52:41
%S A192773 0,1,0,4,3,18,30,98,219,596,1464,3783,9540,24328,61740,156985,398904,
%T A192773 1013772,2576475,6547574,16640382,42288806,107473443,273129468,
%U A192773 694130016,1764047839,4483130424,11393354512,28954911624,73585574049
%N A192773 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.
%C A192773 For discussions of polynomial reduction, see A192232 and A192744.
%H A192773 Harvey P. Dale, <a href="/A192773/b192773.txt">Table of n, a(n) for n = 1..1000</a>
%H A192773 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-1,-5,1,1).
%F A192773 a(n) = a(n-1)+5*a(n-2)-a(n-3)-5*a(n-4)+a(n-5)+a(n-6).
%F A192773 G.f.: x^2*(x^2+x-1)/(x^6+x^5-5*x^4-x^3+5*x^2+x-1). [_Colin Barker_, Nov 23 2012]
%e A192773 The first five polynomials p(n,x) and their reductions are as follows:
%e A192773 F1(x)=1 -> 1
%e A192773 F2(x)=x -> x
%e A192773 F3(x)=x^2+1 -> x^2+1
%e A192773 F4(x)=x^3+2x -> x^2+4x+1
%e A192773 F5(x)=x^4+3x^2+1 -> 6x^2+3x+2, so that
%e A192773 A192772=(1,0,1,1,2,...), A192773=(0,1,0,4,3,...), A192774=(0,0,1,1,6,...)
%t A192773 (See A192772.)
%t A192773 LinearRecurrence[{1,5,-1,-5,1,1},{0,1,0,4,3,18},40] (* _Harvey P. Dale_, Aug 07 2025 *)
%Y A192773 Cf. A192232, A192744, A192772.
%K A192773 nonn,easy
%O A192773 1,4
%A A192773 _Clark Kimberling_, Jul 09 2011