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%I #28 Aug 21 2020 14:33:56
%S 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,
%T 2,3015,2,1,8058,3,182,22151,2,3,61326,411,2,170883,2,11,479410,3,2,
%U 1345211,158,2955,3798246,11,2,10761723,8078,411,30585834,3,2,87191759,2,3,249057230,1,61346,713205963,2,11,2046590850,173775,2
%N 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.
%C a(2^k)=1 for all k including k=0.
%C a(p)=2, a(2p)=3, a(4p)=11, etc. for an odd prime p.
%H Antti Karttunen, <a href="/A115116/b115116.txt">Table of n, a(n) for n = 1..6300</a>
%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) = A115114(n) - A006575(n).
%t A006575[n_] := DivisorSum[n, If[BitAnd[#, 1] == 1, MoebiusMu[#] (3^(n/#) - 1), 0]&]/(2n);
%t A115114[n_] := Sum[EulerPhi[2d] + Boole[OddQ[d]] EulerPhi[d] 3^(n/d), {d, Divisors[n]}]/(2n);
%t a[n_] := A115114[n] - A006575[n];
%t Array[a, 60] (* _Jean-François Alcover_, Aug 29 2019 *)
%o (PARI)
%o A006575(n) = (sumdiv(n,d,bitand(d,1)*moebius(d)*(3^(n/d)-1)) / (2*n)); \\ From A006575.
%o A115114(n) = (1/(2*n))*(sumdiv(n,d,eulerphi(2*d)+(bitand(d,1)*eulerphi(d)*(3^(n/d)))));
%o A115116(n) = (A115114(n) - A006575(n)); \\ _Antti Karttunen_, Jan 19 2020
%Y Cf. A006575, A115114.
%K easy,nonn
%O 1,3
%A _Valery A. Liskovets_, Jan 17 2006
%E More terms from _Antti Karttunen_, Jan 19 2020
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