A192777 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1. See Comments.

1, 0, 1, 1, 2, 8, 14, 55, 121, 392, 989, 2912, 7797, 22104, 60553, 169289, 467622, 1300888, 3603914, 10008543, 27755249, 77034176, 213702153, 593005504, 1645265209, 4565154816, 12666317073, 35144684065, 97512548090, 270561677224

Offset

1,5

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (1,6,-1,-6,1,1).

Formula

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

G.f.: -x*(1-5*x^2+x^4-x+x^3) / ( (x^2-x-1)*(x^4+2*x^3-3*x^2-2*x+1) ). - R. J. Mathar, May 06 2014

Example

The first five polynomials p(n,x) and their reductions are as follows:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+5x+1
F5(x)=x^4+3x^2+1 -> 7x^2+4x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,5,4,...), A192779=(0,0,1,1,7,...)

Mathematica

q = x^3; s = x^2 + 3 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}]
  (* A192777 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192778 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192779 *)

Crossrefs

Cf. A192744, A192232, A192616, A192772, A192778, A192779.

Sequence in context: A111001 A055258 A277649 * A054981 A059449 A272930

Adjacent sequences: A192774 A192775 A192776 * A192778 A192779 A192780

Keyword

nonn

Author

Clark Kimberling, Jul 09 2011

Status

approved