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A115117
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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.
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0
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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
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OFFSET
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1,2
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LINKS
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Table of n, a(n) for n=1..25.
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.
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FORMULA
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a(n) = (Sum_{d|n} phi(3d) + Sum_{d|n, (3,d)=1} mu(d) 4^(n/d))/(3n), where mu(n) is the Moebius function A008683.
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CROSSREFS
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Cf. A006575, A115115.
Sequence in context: A000150 A318232 A304787 * A029890 A095268 A118397
Adjacent sequences: A115114 A115115 A115116 * A115118 A115119 A115120
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KEYWORD
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easy,nonn
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AUTHOR
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Valery A. Liskovets, Jan 17 2006
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STATUS
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approved
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