A046209 Number of ternary Lyndon words whose digits sum to 0 mod 3; also number of trace 0 irreducible polynomials over GF(3).

1, 1, 2, 6, 16, 38, 104, 270, 726, 1960, 5368, 14736, 40880, 113828, 318848, 896670, 2532160, 7174050, 20390552, 58112088, 166037248, 475467916, 1364393896, 3922624800, 11297181456, 32588003000, 94143178098, 272342710380, 788854912240, 2287679084096, 6641649422408, 19302293185470

Offset

1,3

Comments

Also number of ternary Lyndon words of trace 0 over GF(3).

Links

G. C. Greubel, Table of n, a(n) for n = 1..2000

F. Ruskey, Number of q-ary Lyndon words with given trace mod q

F. Ruskey, Number of monic irreducible polynomials over GF(q) with given trace

F. Ruskey, Number of Lyndon words over GF(q) with given trace

Index entries for sequences related to Lyndon words

Formula

a(n) = 1/(3*n) * sum(d divides n, gcd(d, 3)*mu(d)*3^(n/d) ).

a(n) = A053548(n) + A053560(n) + A053561(n). - R. J. Mathar, Oct 21 2021

Example

a(4) = 6 = |{ 0012, 0021, 0111, 0102, 0222, 1122 }|.

Mathematica

a[n_] := 1/(3n) DivisorSum[n, GCD[#, 3]*MoebiusMu[#]*3^(n/#)&]; Array[a, 32] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

Prog

(PARI) a(n) = 1/(3*n) * sumdiv(n, d, gcd(d, 3)*moebius(d)*3^(n/d) ); /* Joerg Arndt, Aug 17 2012 */

Crossrefs

Cf. A046211.

Sequence in context: A265758 A265107 A217631 * A285885 A273348 A198951

Adjacent sequences: A046206 A046207 A046208 * A046210 A046211 A046212

Keyword

nonn

Author

Frank Ruskey, Dec 13 1999

Status

approved