A207873 Numerator of Z(n,1/2), where Z(n,x) is the n-th Zeckendorf polynomial.

1, 1, 1, 5, 1, 9, 5, 1, 17, 9, 5, 21, 1, 33, 17, 9, 41, 5, 37, 21, 1, 65, 33, 17, 81, 9, 73, 41, 5, 69, 37, 21, 85, 1, 129, 65, 33, 161, 17, 145, 81, 9, 137, 73, 41, 169, 5, 133, 69, 37, 165, 21, 149, 85, 1, 257, 129, 65, 321, 33, 289, 161, 17, 273, 145, 81, 337

Offset: 1

Author

Clark Kimberling, Feb 21 2012

Keywords

• nonn

Comments

The Zeckendorf polynomials Z(x,n) are defined and ordered at A207813. See A207872 for denominators to match A207873.

Crossrefs

Cf. A207813, A207873.

Programs

Mathematica

fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k],
    AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],
      {n, 1, 500}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 40}]
Denominator[Table[p[n, x] /. x -> 1/2,
   {n, 1, 120}]]                       (* A207872 *)
Numerator[Table[p[n, x] /. x -> 1/2,
   {n, 1, 120}]]                       (* A207873 *)