A115114 - OEIS

A115114

Asymmetric rhythm cycles (patterns): 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.

%I #25 Oct 27 2024 04:48:16

%S 2,3,6,11,26,63,158,411,1098,2955,8054,22151,61322,170823,478318,

%T 1345211,3798242,10761723,30585830,87169619,249056138,713205903,

%U 2046590846,5883948951,16945772210,48882035163,141214768974

%N Asymmetric rhythm cycles (patterns): 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.

%H R. W. Hall and P. Klingsberg, <a href="https://archive.bridgesmathart.org/2004/bridges2004-189.html">Asymmetric Rhythms, Tiling Canons and Burnside's Lemma</a>, Bridges Proceedings, pp. 189-194, 2004 (Winfield, Kansas).

%H R. W. Hall and P. Klingsberg, <a href="https://doi.org/10.1080/00029890.2006.11920376">Asymmetric Rhythms and Tiling Canons</a>, Preprint, 2004; The American Mathematical Monthly, Volume 113, 2006 - Issue 10, [<a href="https://www.jstor.org/stable/27642087">alternative link</a>].

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

%F a(n) ~ 3^n/(2*n). - _Vaclav Kotesovec_, Oct 27 2024

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

%t a[n_] := Sum[EulerPhi[2d] + Boole[OddQ[d]] EulerPhi[d] 3^(n/d), {d, Divisors[n]}]/(2n);

%t Array[a, 27] (* _Jean-François Alcover_, Aug 29 2019 *)

%Y Cf. A000016, A006575.

%K easy,nonn

%O 1,1

%A _Valery A. Liskovets_, Jan 17 2006