A192616 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1. See Comments.

1, 0, 1, 1, 2, 6, 10, 29, 57, 142, 309, 720, 1625, 3714, 8457, 19259, 43902, 99970, 227830, 518943, 1182401, 2693624, 6136837, 13980960, 31851853, 72565704, 165320833, 376638417, 858066430, 1954869262, 4453630790, 10146374277, 23115721705

Offset: 1

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn
• easy

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+3x+1
F5(x)=x^4+3x^2+1 -> 4x^2+2x+2, so that
A192616=(1,0,1,1,2,...), A192617=(0,1,0,3,2,...), A192651=(0,0,1,1,5,...)
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-4,1,1).

Crossrefs

Cf. A192744, A192232, A192617, A192651.

Programs

Mathematica

q = x^3; s = x^2 + x + 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}]
  (* A192616 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192617 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192651 *)

Formula

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

G.f.: -x*(x^4+x^3-3*x^2-x+1)/(x^6+x^5-4*x^4-x^3+4*x^2+x-1). [Colin Barker, Jul 27 2012]