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

1,3

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].

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A006575, A115114.

Sequence in context: A171565 A328266 A359895 * A141662 A328383 A088062

Adjacent sequences: A115113 A115114 A115115 * A115117 A115118 A115119

Keyword

easy,nonn

Author

Valery A. Liskovets, Jan 17 2006

Extensions

More terms from Antti Karttunen, Jan 19 2020

Status

approved