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A207813 Numbers that match irreducible Zeckendorf polynomials. 7
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 15 10:54 EDT 2021. Contains 343909 sequences. (Running on oeis4.)