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A006575
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Number of primitive (aperiodic, or Lyndon) asymmetric rhythm cycles: ones having no nontrivial shift automorphism.
(Formerly M1204)
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7
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1, 2, 4, 10, 24, 60, 156, 410, 1092, 2952, 8052, 22140, 61320, 170820, 478288, 1345210, 3798240, 10761660, 30585828, 87169608, 249055976, 713205900, 2046590844, 5883948540, 16945772184, 48882035160, 141214767876
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OFFSET
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1,2
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COMMENTS
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Asymmetric rhythm cycles (A115114): binary necklaces of length 2n subject to the restriction that for any k if the k-th bead is of color 1 then the (k+n)-th bead (modulo 2n) is of color 0. - Valery A. Liskovets, Jan 17 2006
This sequence differs from the Moebius transform of A115114 (for even n). Coincides with the second row (q=3) of array A098691. - Valery A. Liskovets, Jan 17 2006
This sequence is the number of Lyndon words on {1, 2, 3} with an odd number of 1's. Also, for even n, this sequence represents the differences between the number of Lyndon words on {1, 2, 3} with an odd number of 1's and the number of Lyndon words on {1, 2, 3} with an even number of 1's. - Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Jan 03 2008
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Shanks and M. Lal, Bateman's constants reconsidered and the distribution of cubic residues, Math. Comp., 26 (1972), 265-285.
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LINKS
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Joerg Arndt, Table of n, a(n) for n = 1..200
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, d odd}mu(d)(3^(n/d)-1))/(2n). Also a(n)=(3^n-1)/(2n) for n=2^k and a(n)=(Sum_{d|n, d odd}mu(d)3^(n/d))/(2n) otherwise. - Valery A. Liskovets, Jan 17 2006
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EXAMPLE
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Example. For n=3, out of 6=A115114(3) admissible rhythm cycles (necklaces) 000000, 100000, 110000, 101000, 111000 and 101010, only the first and the last ones are imprimitive. Thus a(3)=4.
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PROG
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(PARI) a(n) = sumdiv( n, d, if ( bitand(d, 1), moebius(d) * (3^(n/d)-1) , 0 ) ) / (2*n); /* Joerg Arndt, Dec 30 2012 */
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CROSSREFS
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Cf. A133267.
Sequence in context: A100087 A088354 A055919 * A138175 A121691 A124499
Adjacent sequences: A006572 A006573 A006574 * A006576 A006577 A006578
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Edited and extended by Valery A. Liskovets, Jan 17 2006
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STATUS
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approved
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