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

%I #6 Jul 12 2012 00:40:00

%S 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,

%T 5,69,37,21,85,1,129,65,33,161,17,145,81,9,137,73,41,169,5,133,69,37,

%U 165,21,149,85,1,257,129,65,321,33,289,161,17,273,145,81,337

%N Numerator 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. See A207872 for denominators to match A207873.

%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

%O 1,4

%A _Clark Kimberling_, Feb 21 2012