A115114 -id:A115114 - cp's OEIS frontend

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

Valery A. Liskovets, Jan 17 2006

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.

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].

Cf. A006575, A115114.

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 *)

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

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

More terms from Antti Karttunen, Jan 19 2020

A006575 Number of primitive (aperiodic, or Lyndon) asymmetric rhythm cycles: ones having no nontrivial shift automorphism.

Original entry on oeis.org

1, 2, 4, 10, 24, 60, 156, 410, 1092, 2952, 8052, 22140, 61320, 170820, 478288, 1345210, 3798240, 10761660, 30585828, 87169608, 249055976, 713205900, 2046590844, 5883948540, 16945772184, 48882035160, 141214767876

Offset: 1

N. J. A. Sloane

nonn

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. - Valery A. Liskovets, Jan 17 2006
This sequence differs from the Moebius transform of A115114 (for even n). Coincides with the second row (q=3) of array A098691. - Valery A. Liskovets, Jan 17 2006
This sequence is the number of Lyndon words on {1, 2, 3} with an odd number of 1's. Also, for even n, this sequence represents the differences between the number of Lyndon words on {1, 2, 3} with an odd number of 1's and the number of Lyndon words on {1, 2, 3} with an even number of 1's. - Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Jan 03 2008

			Example. For n=3, out of 6=A115114(3) admissible rhythm cycles (necklaces) 000000, 100000, 110000, 101000, 111000 and 101010, only the first and the last ones are imprimitive. Thus a(3)=4.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joerg Arndt, Table of n, a(n) for n = 1..200
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].
D. Shanks and M. Lal, Bateman's constants reconsidered and the distribution of cubic residues, Math. Comp., 26 (1972), 265-285.

Cf. A133267. Row q=3 of A098691.

a[n_] := DivisorSum[n, If[BitAnd[#, 1]==1, MoebiusMu[#]*(3^(n/#)-1), 0]&] / (2n); Array[a, 30] (* Jean-François Alcover, Dec 01 2015, after Joerg Arndt *)

a(n) = sumdiv( n, d, if ( bitand(d,1), moebius(d) * (3^(n/d)-1) , 0 ) ) / (2*n); /* Joerg Arndt, Dec 30 2012 */

From Valery A. Liskovets, Jan 17 2006: (Start)
a(n) = (Sum_{d|n, d odd} mu(d)*(3^(n/d)-1))/(2*n).
a(n) = (3^n-1)/(2*n) for n=2^k and a(n) = (Sum_{d|n, d odd} mu(d)*3^(n/d))/(2*n) otherwise. (End)

Edited and extended by Valery A. Liskovets, Jan 17 2006

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.

Original entry on oeis.org

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

Valery A. Liskovets, Jan 17 2006

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].

Cf. A115114.

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 *)

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

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

cp's OEIS Frontend

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.

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A006575 Number of primitive (aperiodic, or Lyndon) asymmetric rhythm cycles: ones having no nontrivial shift automorphism.

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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.

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