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

%I A207872 #24 Sep 06 2020 03:45:26
%S A207872 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 A207872 64,64,64,64,64,64,64,64,128,128,128,128,128,128,128,128,128,128,128,
%U A207872 128,128,128,128,128,128,128,128,128,128,256,256,256,256,256
%N A207872 Denominator of Z(n,1/2), where Z(n,x) is the n-th Zeckendorf polynomial.
%C A207872 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 A207872 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 A207872 Sajed Haque, <a href="http://hdl.handle.net/10012/12234">Discriminators of Integer Sequences</a>, Thesis, 2017, See p. 36.
%t A207872 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 A207872     AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],
%t A207872       {n, 1, 500}];
%t A207872 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A207872 p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A207872 Table[p[n, x], {n, 1, 40}]
%t A207872 Denominator[Table[p[n, x] /. x -> 1/2,
%t A207872    {n, 1, 120}]]                       (* A207872 *)
%t A207872 Numerator[Table[p[n, x] /. x -> 1/2,
%t A207872    {n, 1, 120}]]                       (* A207873 *)
%Y A207872 Cf. A207813, A207873.
%K A207872 nonn,frac
%O A207872 1,2
%A A207872 _Clark Kimberling_, Feb 21 2012