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A192780 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+1. See Comments.

%I A192780 #10 Nov 07 2021 18:02:17
%S A192780 1,0,1,1,2,5,8,19,34,71,137,272,537,1056,2089,4112,8121,16009,31586,
%T A192780 62301,122888,242411,478146,943183,1860433,3669792,7238769,14278720,
%U A192780 28165265,55556896,109587889,216165713,426394178,841076725,1659052040
%N A192780 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+1. See Comments.
%C A192780 For discussions of polynomial reduction, see A192232 and A192744.
%H A192780 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,-3,1,1).
%F A192780 a(n)=a(n-1)+3*a(n-2)-a(n-3)-3*a(n-4)+a(n-5)+a(n-6).
%F A192780 G.f.: -x*(x-1)*(1+x)*(x^2+x-1) / ( -1+x+3*x^2-x^3-3*x^4+x^5+x^6 ). - _R. J. Mathar_, May 06 2014
%e A192780 The first five polynomials p(n,x) and their reductions:
%e A192780 F1(x)=1 -> 1
%e A192780 F2(x)=x -> x
%e A192780 F3(x)=x^2+1 -> x^2+1
%e A192780 F4(x)=x^3+2x -> x^2+2x+1
%e A192780 F5(x)=x^4+3x^2+1 -> 4x^2+1x+2, so that
%e A192780 A192777=(1,0,1,1,2,...), A192778=(0,1,0,2,1,...), A192779=(0,0,1,1,4,...)
%t A192780 q = x^3; s = x^2 + 1; z = 40;
%t A192780 p[n_, x_] := Fibonacci[n, x];
%t A192780 Table[Expand[p[n, x]], {n, 1, 7}]
%t A192780 reduce[{p1_, q_, s_, x_}] :=
%t A192780 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192780        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192780 t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];
%t A192780 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192780   (* A192780 *)
%t A192780 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192780   (* A192781 *)
%t A192780 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
%t A192780   (* A192782 *)
%t A192780 LinearRecurrence[{1,3,-1,-3,1,1},{1,0,1,1,2,5},40] (* _Harvey P. Dale_, Nov 07 2021 *)
%Y A192780 Cf. A192744, A192232, A192616, A192781, A192782.
%K A192780 nonn
%O A192780 1,5
%A A192780 _Clark Kimberling_, Jul 09 2011