A207813 Numbers that match irreducible Zeckendorf polynomials.

2, 4, 9, 17, 19, 25, 27, 30, 40, 43, 46, 53, 56, 59, 61, 67, 69, 72, 77, 82, 85, 93, 95, 98, 101, 103, 108, 111, 114, 119, 124, 129, 135, 137, 140, 150, 153, 161, 166, 169, 171, 177, 179, 182, 187, 195, 197, 205, 208, 211, 213, 218, 224, 229, 237, 239

Offset

1,1

Comments

The Zeckendorf representation of a positive integer n is

a unique sum

...

c(k-2)F(k) + c(k-3)F(k-1) + ... + c(1)F(3) + c(0)F(2),

...

where F=A000045 (Fibonacci numbers), c(k-2)=1, and for

j=0,1,...,k-3, there are two restrictions on

coefficients: c(j) is 0 or 1, and c(j)c(j+1)=0; viz.,

no two consecutive Fibonacci numbers appear. The

Zeckendorf polynomial Z(n,x) is introduced here as

...

c(k-2)x^(k-2) + c(k-3)x^(k-3) + ... + c(1)x + c(0) .

...

A207813 refers to irreduciblity over the field of

rational numbers.

Links

Table of n, a(n) for n=1..56.

Example

n...k...Z(n)....Z(n,x).......irreducible
1...2...1.......1............no
2...3...10......x............yes
3...4...100.....x^2..........no
4...4...101.....x^2 + 1......yes
5...5...1000....x^3..........no
6...5...1001....x^3 + 1......no
7...5...1010....x^3 + x .....no
8...5...10000...x^4..........no
9...5...10001...x^4 + 1......yes

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, 350}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 40}] (* Zeckendorf polynomials *)
u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
  AppendTo[u, n]], {n, 300}]; u     (* A207813 *)

Crossrefs

Cf. A206073, A206074.

Sequence in context: A060401 A063981 A131095 * A136379 A065026 A199205

Adjacent sequences: A207810 A207811 A207812 * A207814 A207815 A207816

Keyword

nonn

Author

Clark Kimberling, Feb 20 2012

Status

approved