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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
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
Sequence in context: A060401 A063981 A131095 * A136379 A065026 A199205
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 20 2012
STATUS
approved

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Last modified April 18 06:24 EDT 2024. Contains 371769 sequences. (Running on oeis4.)