cp's OEIS Frontend

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

%I A207822 #7 Mar 30 2012 18:58:13
%S A207822 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 A207822 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 A207822 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 A207822 Number of distinct irreducible factors of n-th Zeckendorf polynomial.
%C A207822 The Zeckendorf polynomials Z(x,n) are defined and ordered at A207813.
%e A207822 Z(10,n) = x^4 + x = x(x + 1)(x^2 - x + 1), so a(10)=3.
%t A207822 fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],
%t A207822    t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],
%t A207822     AppendTo[fr, 1]; t = t - Fibonacci[k],
%t A207822   AppendTo[fr, 0]]; k--]; fr];
%t A207822 t = Table[fb[n], {n, 1, 500}];
%t A207822 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A207822 p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A207822 TableForm[Table[{n, p[n, x], FactorList[p[n, x]]},
%t A207822 {n, 1, 10}]]
%t A207822 Table[-1 + Length[FactorList[p[n, x]]], {n, 1, 120}]
%Y A207822 Cf. A207813.
%K A207822 nonn
%O A207822 1,6
%A A207822 _Clark Kimberling_, Feb 20 2012