A207872 - OEIS

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

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OFFSET

1,2

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

Table of n, a(n) for n=1..59.

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

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 *)