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A207872 Denominator of Z(n,1/2), where Z(n,x) is the n-th Zeckendorf polynomial. 5

%I

%S 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,

%T 64,64,64,64,64,64,64,64,128,128,128,128,128,128,128,128,128,128,128,

%U 128,128,128,128,128,128,128,128,128,128,256,256,256,256,256

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

%C 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).

%C 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

%H Sajed Haque, <a href="http://hdl.handle.net/10012/12234">Discriminators of Integer Sequences</a>, Thesis, 2017, See p. 36.

%t 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],

%t AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],

%t {n, 1, 500}];

%t b[n_] := Reverse[Table[x^k, {k, 0, n}]]

%t p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]

%t Table[p[n, x], {n, 1, 40}]

%t Denominator[Table[p[n, x] /. x -> 1/2,

%t {n, 1, 120}]] (* A207872 *)

%t Numerator[Table[p[n, x] /. x -> 1/2,

%t {n, 1, 120}]] (* A207873 *)

%Y Cf. A207813, A207873.

%K nonn,frac

%O 1,2

%A _Clark Kimberling_, Feb 21 2012

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Last modified June 21 03:24 EDT 2021. Contains 345354 sequences. (Running on oeis4.)