

A115114


Asymmetric rhythm cycles (patterns): binary necklaces of length 2n subject to the restriction that for any k if the kth bead is of color 1 then the (k+n)th bead (modulo 2n) is of color 0.


3



2, 3, 6, 11, 26, 63, 158, 411, 1098, 2955, 8054, 22151, 61322, 170823, 478318, 1345211, 3798242, 10761723, 30585830, 87169619, 249056138, 713205903, 2046590846, 5883948951, 16945772210, 48882035163, 141214768974
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OFFSET

1,1


LINKS

Table of n, a(n) for n=1..27.
R. W. Hall and P. Klingsberg, Asymmetric Rhythms, Tiling Canons and Burnside's Lemma,Bridges Proceedings, pp. 189194, 2004 (Winfield, Kansas).
R. W. Hall and P. Klingsberg, Asymmetric Rhythms and Tiling Canons, Preprint, 2004.


FORMULA

a(n)=(Sum_{dn}phi(2d)+Sum_{dn, d odd}phi(d)3^(n/d))/(2n), where phi(n) is the Euler function A000010.


EXAMPLE

For n=3, the 27=3^3 admissible words are separated into 6 shiftequivalence classes (necklaces) containing, resp., the words 000000, 100000, 110000, 101000, 111000 and 101010. Thus a(3)=6.


CROSSREFS

Cf. A000016, A006575.
Sequence in context: A051603 A094927 A024423 * A086209 A022490 A102952
Adjacent sequences: A115111 A115112 A115113 * A115115 A115116 A115117


KEYWORD

easy,nonn


AUTHOR

Valery A. Liskovets, Jan 17 2006


STATUS

approved



