A115117 Number of primitive (aperiodic, or Lyndon) 3-asymmetric rhythm cycles: ones having no nontrivial shift automorphism. 3-asymmetric rhythm cycles (A115115): binary necklaces of length 3n subject to the restriction that for any k if the k-th bead is of color 1 then the (k+n)-th and (k+2n)-th beads (modulo 3n) are of color 0.

1, 2, 7, 20, 68, 224, 780, 2720, 9709, 34918, 127100, 465920, 1720740, 6390930, 23860928, 89477120, 336860180, 1272578048, 4822419420, 18325176316, 69810262080, 266548209850, 1019836872140, 3909374443520, 15011998757888

Offset

1,2

Links

Jinyuan Wang, Table of n, a(n) for n = 1..1000

R. W. Hall and P. Klingsberg, Asymmetric Rhythms, Tiling Canons and Burnside's Lemma, Bridges Proceedings, pp. 189-194, 2004 (Winfield, Kansas).

R. W. Hall and P. Klingsberg, Asymmetric Rhythms and Tiling Canons, Preprint, 2004.

Formula

a(n) = (Sum_{d|n} mu(3d) + Sum_{d|n, (3,d)=1} mu(d) 4^(n/d))/(3n), where mu(n) is the Moebius function A008683.

a(n) ~ 4^n / (3*n). - Vaclav Kotesovec, Oct 27 2024

Mathematica

a[n_] := Sum[MoebiusMu[3d] + Boole[GCD[3, d] == 1] MoebiusMu[d] 4^(n/d), {d, Divisors[n]}]/(3n);
Array[a, 25] (* Jean-François Alcover, Aug 30 2019 *)

Prog

(PARI) a(n) = 1/(3*n) * sumdiv(n, d, moebius(3*d) + if(gcd(3, d)==1, moebius(d)*4^(n/d), 0) ); \\ Joerg Arndt, Aug 29 2019

Crossrefs

Cf. A006575, A115115.

Sequence in context: A000150 A318232 A304787 * A029890 A095268 A118397

Adjacent sequences: A115114 A115115 A115116 * A115118 A115119 A115120

Keyword

easy,nonn

Author

Valery A. Liskovets, Jan 17 2006

Status

approved