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%I #27 Aug 21 2020 12:11:14
%S 1,2,7,20,68,224,780,2720,9709,34918,127100,465920,1720740,6390930,
%T 23860928,89477120,336860180,1272578048,4822419420,18325176316,
%U 69810262080,266548209850,1019836872140,3909374443520,15011998757888
%N 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.
%H Jinyuan Wang, <a href="/A115117/b115117.txt">Table of n, a(n) for n = 1..1000</a>
%H R. W. Hall and P. Klingsberg, <a href="https://www.semanticscholar.org/paper/Asymmetric-Rhythms%2C-Tiling-Canons%2C-and-Burnside's-Klingsberg/01d101fc4ef7ce5f1f9eeeeeafb046ede9cb4ea8">Asymmetric Rhythms, Tiling Canons and Burnside's Lemma</a>, Bridges Proceedings, pp. 189-194, 2004 (Winfield, Kansas).
%H R. W. Hall and P. Klingsberg, <a href="http://www.sju.edu/%7Erhall/Rhythms/AsymmetricRhythms/canons.pdf">Asymmetric Rhythms and Tiling Canons</a>, Preprint, 2004.
%F 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.
%t a[n_] := Sum[MoebiusMu[3d] + Boole[GCD[3, d] == 1] MoebiusMu[d] 4^(n/d), {d, Divisors[n]}]/(3n);
%t Array[a, 25] (* _Jean-François Alcover_, Aug 30 2019 *)
%o (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
%Y Cf. A006575, A115115.
%K easy,nonn
%O 1,2
%A _Valery A. Liskovets_, Jan 17 2006
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