login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006575 Number of primitive (aperiodic, or Lyndon) asymmetric rhythm cycles: ones having no nontrivial shift automorphism.
(Formerly M1204)
7
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 (list; graph; refs; listen; history; text; internal format)
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).

D. Shanks and M. Lal, Bateman's constants reconsidered and the distribution of cubic residues, Math. Comp., 26 (1972), 265-285.

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.

FORMULA

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

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-Fran├žois 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.

Sequence in context: A291419 A088354 A055919 * A230551 A246566 A230553

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 20 12:31 EDT 2017. Contains 292271 sequences.