A006575 Number of primitive (aperiodic, or Lyndon) asymmetric rhythm cycles: ones having no nontrivial shift automorphism.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

			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.
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• 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; The American Mathematical Monthly, Volume 113, 2006 - Issue 10, [alternative link].
• D. Shanks and M. Lal, Bateman's constants reconsidered and the distribution of cubic residues, Math. Comp., 26 (1972), 265-285.

Crossrefs

Cf. A133267. Row q=3 of A098691.

Programs

Mathematica

a[n_] := DivisorSum[n, If[BitAnd[#, 1]==1, MoebiusMu[#]*(3^(n/#)-1), 0]&] / (2n); Array[a, 30] (* Jean-François Alcover, Dec 01 2015, after Joerg Arndt *)

PARI

a(n) = sumdiv( n, d, if ( bitand(d,1), moebius(d) * (3^(n/d)-1) , 0 ) ) / (2*n); /* Joerg Arndt, Dec 30 2012 */

Formula

From Valery A. Liskovets, Jan 17 2006: (Start)

a(n) = (Sum_{d|n, d odd} mu(d)*(3^(n/d)-1))/(2*n).

a(n) = (3^n-1)/(2*n) for n=2^k and a(n) = (Sum_{d|n, d odd} mu(d)*3^(n/d))/(2*n) otherwise. (End)

Extensions

Edited and extended by Valery A. Liskovets, Jan 17 2006