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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, 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
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
Sequence in context: A171565 A328266 A359895 * A141662 A328383 A088062
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Jan 17 2006
EXTENSIONS
More terms from Antti Karttunen, Jan 19 2020
STATUS
approved