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

1, 0, 1, 1, 2, 5, 8, 19, 34, 71, 137, 272, 537, 1056, 2089, 4112, 8121, 16009, 31586, 62301, 122888, 242411, 478146, 943183, 1860433, 3669792, 7238769, 14278720, 28165265, 55556896, 109587889, 216165713, 426394178, 841076725, 1659052040

Offset: 1

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Examples

			The first five polynomials p(n,x) and their reductions:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+2x+1
F5(x)=x^4+3x^2+1 -> 4x^2+1x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,2,1,...), A192779=(0,0,1,1,4,...)
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-3,1,1).

Crossrefs

Cf. A192744, A192232, A192616, A192781, A192782.

Programs

Mathematica

q = x^3; s = x^2 + 1; z = 40;
p[n_, x_] := Fibonacci[n, x];
Table[Expand[p[n, x]], {n, 1, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192780 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192781 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192782 *)
LinearRecurrence[{1,3,-1,-3,1,1},{1,0,1,1,2,5},40] (* Harvey P. Dale, Nov 07 2021 *)

Formula

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

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