|
| |
|
|
A115117
|
|
Number of primitive (aperiodic, or Lyndon) 3-asymmetric rhythm cycles: ones having no nontrivial shift automorphism. 3-asymmetric rhythm cycles (A115115): binary necklaces of length 3n subject to the restriction that for any k if the k-th bead is of color 1 then the (k+n)-th and (k+2n)-th beads (modulo 3n) are of color 0.
|
|
1
|
|
|
|
1, 2, 7, 20, 68, 224, 780, 2720, 9709, 34918, 127100, 465920, 1720740, 6390930, 23860928, 89477120, 336860180, 1272578048, 4822419420, 18325176316, 69810262080, 266548209850, 1019836872140, 3909374443520, 15011998757888
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
Jinyuan Wang, Table of n, a(n) for n = 1..1000
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} mu(3d) + Sum_{d|n, (3,d)=1} mu(d) 4^(n/d))/(3n), where mu(n) is the Moebius function A008683.
|
|
|
MATHEMATICA
|
a[n_] := Sum[MoebiusMu[3d] + Boole[GCD[3, d] == 1] MoebiusMu[d] 4^(n/d), {d, Divisors[n]}]/(3n);
Array[a, 25] (* Jean-François Alcover, Aug 30 2019 *)
|
|
|
PROG
|
(PARI) a(n) = 1/(3*n) * sumdiv(n, d, moebius(3*d) + if(gcd(3, d)==1, moebius(d)*4^(n/d), 0) ); \\ Joerg Arndt, Aug 29 2019
|
|
|
CROSSREFS
|
Cf. A006575, A115115.
Sequence in context: A000150 A318232 A304787 * A029890 A095268 A118397
Adjacent sequences: A115114 A115115 A115116 * A115118 A115119 A115120
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
Valery A. Liskovets, Jan 17 2006
|
|
|
STATUS
|
approved
|
| |
|
|
