A006575 Number of primitive (aperiodic, or Lyndon) asymmetric rhythm cycles: ones having no nontrivial shift automorphism. (Formerly M1204)

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

1,2

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

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.

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)

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A133267. Row q=3 of A098691.

Sequence in context: A088354 A329673 A055919 * A377256 A307900 A230551

Adjacent sequences: A006572 A006573 A006574 * A006576 A006577 A006578

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved