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

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, 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, 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

Offset: 1

Author

Clark Kimberling, Feb 20 2012

Keywords

• nonn

Comments

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

Examples

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

Crossrefs

Cf. A207813.

Programs

Mathematica

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],
  AppendTo[fr, 0]]; k--]; fr];
t = Table[fb[n], {n, 1, 500}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x], FactorList[p[n, x]]},
{n, 1, 10}]]
Table[-1 + Length[FactorList[p[n, x]]], {n, 1, 120}]