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%I M1204 #40 Dec 26 2021 21:36:57
%S 1,2,4,10,24,60,156,410,1092,2952,8052,22140,61320,170820,478288,
%T 1345210,3798240,10761660,30585828,87169608,249055976,713205900,
%U 2046590844,5883948540,16945772184,48882035160,141214767876
%N Number of primitive (aperiodic, or Lyndon) asymmetric rhythm cycles: ones having no nontrivial shift automorphism.
%C 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
%C 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
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Joerg Arndt, <a href="/A006575/b006575.txt">Table of n, a(n) for n = 1..200</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>].
%H D. Shanks and M. Lal, <a href="https://doi.org/10.1090/S0025-5718-1972-0302590-7">Bateman's constants reconsidered and the distribution of cubic residues</a>, Math. Comp., 26 (1972), 265-285.
%F From _Valery A. Liskovets_, Jan 17 2006: (Start)
%F a(n) = (Sum_{d|n, d odd} mu(d)*(3^(n/d)-1))/(2*n).
%F 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)
%e 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.
%t 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_ *)
%o (PARI) a(n) = sumdiv( n, d, if ( bitand(d,1), moebius(d) * (3^(n/d)-1) , 0 ) ) / (2*n); /* _Joerg Arndt_, Dec 30 2012 */
%Y Cf. A133267. Row q=3 of A098691.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E Edited and extended by _Valery A. Liskovets_, Jan 17 2006
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