A207872 Denominator of Z(n,1/2), where Z(n,x) is the n-th Zeckendorf polynomial.

1, 2, 4, 4, 8, 8, 8, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256

Offset: 1

Author

Clark Kimberling, Feb 21 2012

Keywords

• nonn
• frac

Comments

The Zeckendorf polynomials Z(x,n) are defined and ordered at A207813. Each power 2^k appears F(k+1) times, where F=A000045 (Fibonacci numbers).

Conjecture: a(n) is also the reverse binarization of the Zeckendorf representation of n in base Fibonacci. For example, 11 = 1x8 + 0x5 +1x3 +0x2 + 0x1, so 11 =10100 in base Fibonacci. Now read that as binary but in reverse, 00101 = 101 = 5 = A207873(11). - George Beck, Sep 02 2020

Links

• Sajed Haque, Discriminators of Integer Sequences, Thesis, 2017, See p. 36.

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 *)