cp's OEIS Frontend

A066313 Number of aperiodic necklaces with n red or blue beads such that two necklaces are equivalent under the operation of simultaneously turning the necklace over and switching the two colors.

%I A066313 #11 Jun 16 2021 15:35:21
%S A066313 1,1,1,2,3,6,9,18,28,57,93,181,315,612,1091,2100,3855,7392,13797,
%T A066313 26436,49929,95790,182361,350440,671088,1292445,2485504,4797261,
%U A066313 9256395,17903316,34636833,67124160,130150493,252675975,490853403,954498874,1857283155,3616938738
%N A066313 Number of aperiodic necklaces with n red or blue beads such that two necklaces are equivalent under the operation of simultaneously turning the necklace over and switching the two colors.
%C A066313 Also number of aperiodic cyclic graphs with oriented edges on n nodes that can be turned over.
%H A066313 Andrew Howroyd, <a href="/A066313/b066313.txt">Table of n, a(n) for n = 1..1000</a>
%H A066313 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A066313 <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F A066313 Moebius transform of A053656.
%e A066313 The equivalence requires the "turning over" operation and the "switching colors" operation to be simultaneous; thus rrrbbrrb is equivalent to rbbrrbbb, but not to bbbrrbbr.
%o A066313 (PARI) \\ here b(n) is A053656.
%o A066313 b(n)={(sumdiv(n, d, eulerphi(d)*2^(n/d))/n + if(n%2==0, 2^(n/2-1)))/2}
%o A066313 a(n)={sumdiv(n, d, moebius(d)*b(n/d))} \\ _Andrew Howroyd_, Jun 14 2021
%Y A066313 Cf. A053656.
%K A066313 nonn
%O A066313 1,4
%A A066313 _Christian G. Bower_, Dec 13 2001; revised Apr 25 2006
%E A066313 Terms a(36) and beyond from _Andrew Howroyd_, Jun 14 2021