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A207822 Number of distinct irreducible factors of n-th Zeckendorf polynomial. 0

%I #7 Mar 30 2012 18:58:13

%S 0,1,1,1,1,2,2,1,1,3,2,2,1,2,2,3,1,2,1,3,1,2,3,2,1,3,1,2,2,1,2,3,2,1,

%T 2,3,3,2,2,1,2,3,1,2,2,1,2,2,2,2,3,3,1,3,1,1,3,3,1,3,1,3,2,3,2,2,1,3,

%U 1,2,2,1,2,3,2,2,1,3,2,2,2,1,4,3,1,2,3,2,1,3,2,3,1,3,1,2,3,1,2

%N Number of distinct irreducible factors of n-th Zeckendorf polynomial.

%C The Zeckendorf polynomials Z(x,n) are defined and ordered at A207813.

%e Z(10,n) = x^4 + x = x(x + 1)(x^2 - x + 1), so a(10)=3.

%t fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],

%t t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],

%t AppendTo[fr, 1]; t = t - Fibonacci[k],

%t AppendTo[fr, 0]]; k--]; fr];

%t t = Table[fb[n], {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 TableForm[Table[{n, p[n, x], FactorList[p[n, x]]},

%t {n, 1, 10}]]

%t Table[-1 + Length[FactorList[p[n, x]]], {n, 1, 120}]

%Y Cf. A207813.

%K nonn

%O 1,6

%A _Clark Kimberling_, Feb 20 2012

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