A115116 Number of imprimitive (periodic) asymmetric rhythm cycles: ones having nontrivial shift automorphisms. 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.

1, 1, 2, 1, 2, 3, 2, 1, 6, 3, 2, 11, 2, 3, 30, 1, 2, 63, 2, 11, 162, 3, 2, 411, 26, 3, 1098, 11, 2, 3015, 2, 1, 8058, 3, 182, 22151, 2, 3, 61326, 411, 2, 170883, 2, 11, 479410, 3, 2, 1345211, 158, 2955, 3798246, 11, 2, 10761723, 8078, 411, 30585834, 3, 2, 87191759, 2, 3, 249057230, 1, 61346, 713205963, 2, 11, 2046590850, 173775, 2

Offset: 1

Author

Valery A. Liskovets, Jan 17 2006

Keywords

• easy
• nonn

Comments

a(2^k)=1 for all k including k=0.

a(p)=2, a(2p)=3, a(4p)=11, etc. for an odd prime p.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..6300
• 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].

Crossrefs

Cf. A006575, A115114.

Programs

Mathematica

A006575[n_] := DivisorSum[n, If[BitAnd[#, 1] == 1, MoebiusMu[#] (3^(n/#) - 1), 0]&]/(2n);
A115114[n_] := Sum[EulerPhi[2d] + Boole[OddQ[d]] EulerPhi[d] 3^(n/d), {d, Divisors[n]}]/(2n);
a[n_] := A115114[n] - A006575[n];
Array[a, 60] (* Jean-François Alcover, Aug 29 2019 *)

PARI

A006575(n) = (sumdiv(n,d,bitand(d,1)*moebius(d)*(3^(n/d)-1)) / (2*n)); \\ From A006575.
A115114(n) = (1/(2*n))*(sumdiv(n,d,eulerphi(2*d)+(bitand(d,1)*eulerphi(d)*(3^(n/d)))));
A115116(n) = (A115114(n) - A006575(n)); \\ Antti Karttunen, Jan 19 2020

Formula

a(n) = A115114(n) - A006575(n).

Extensions

More terms from Antti Karttunen, Jan 19 2020