A115115 Number of 3-asymmetric rhythm cycles: 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.

2, 4, 8, 24, 70, 232, 782, 2744, 9710, 34990, 127102, 466152, 1720742, 6391714, 23860936, 89479864, 336860182, 1272587758, 4822419422, 18325211326, 69810262088, 266548336954, 1019836872142, 3909374909672, 15011998757958

Offset

1,1

Links

Table of n, a(n) for n=1..25.

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].

Formula

a(n) = (Sum_{d|n}phi(3d) + Sum_{d|n, (3, d)=1}phi(d)*4^(n/d))/(3n), where phi(n) is the Euler function A000010.

a(n) ~ 4^n / (3*n). - Vaclav Kotesovec, Aug 28 2019

Mathematica

a[n_] := (Sum[EulerPhi[3d], {d, Divisors[n]}] + Sum[Boole[CoprimeQ[3, d]] EulerPhi[d] 4^(n/d), {d, Divisors[n]}])/(3n);
Array[a, 25] (* Jean-François Alcover, Aug 28 2019 *)

Prog

(PARI) a(n) = (sumdiv(n, d, eulerphi(3*d)) + sumdiv(n, d, if (gcd(d, 3)==1, eulerphi(d)*4^(n/d))))/(3*n); \\ Michel Marcus, Aug 28 2019

Crossrefs

Cf. A115114.

Sequence in context: A231721 A114900 A264570 * A026097 A264557 A067646

Adjacent sequences: A115112 A115113 A115114 * A115116 A115117 A115118

Keyword

easy,nonn

Author

Valery A. Liskovets, Jan 17 2006

Status

approved