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A192772 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.

%I A192772 #13 Feb 19 2025 02:40:46
%S A192772 1,0,1,1,2,7,12,41,86,247,585,1548,3849,9896,25001,63724,161721,
%T A192772 411257,1044878,2655719,6748972,17151849,43589578,110777391,281529169,
%U A192772 715471992,1818293377,4620978640,11743694657,29845241080,75848270001
%N A192772 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.
%C A192772 For discussions of polynomial reduction, see A192232 and A192744.
%H A192772 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-1,-5,1,1).
%F A192772 a(n) = a(n-1)+5*a(n-2)-a(n-3)-5*a(n-4)+a(n-5)+a(n-6).
%F A192772 G.f.: -x*(x^2-x-1)*(x^2+2*x-1) / (x^6+x^5-5*x^4-x^3+5*x^2+x-1). [_Colin Barker_, Jan 17 2013]
%e A192772 The first five polynomials p(n,x) and their reductions are as follows:
%e A192772 F1(x)=1 -> 1
%e A192772 F2(x)=x -> x
%e A192772 F3(x)=x^2+1 -> x^2+1
%e A192772 F4(x)=x^3+2x -> x^2+4x+1
%e A192772 F5(x)=x^4+3x^2+1 -> 6x^2+3x+2, so that
%e A192772 A192772=(1,0,1,1,2,...), A192773=(0,1,0,4,3,...), A192774=(0,0,1,1,6,...)
%t A192772 q = x^3; s = x^2 + 2 x + 1; z = 40;
%t A192772 p[n_, x_] := Fibonacci[n, x];
%t A192772 Table[Expand[p[n, x]], {n, 1, 7}]
%t A192772 reduce[{p1_, q_, s_, x_}] :=
%t A192772 FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192772 t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];
%t A192772 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192772 *)
%t A192772 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192773 *)
%t A192772 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192774 *)
%Y A192772 Cf. A192744, A192232, A192616, A192773, A192774.
%K A192772 nonn,easy
%O A192772 1,5
%A A192772 _Clark Kimberling_, Jul 09 2011