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.

1, 1, 1, 2, 3, 6, 9, 18, 28, 57, 93, 181, 315, 612, 1091, 2100, 3855, 7392, 13797, 26436, 49929, 95790, 182361, 350440, 671088, 1292445, 2485504, 4797261, 9256395, 17903316, 34636833, 67124160, 130150493, 252675975, 490853403, 954498874, 1857283155, 3616938738

Offset: 1

Author

Christian G. Bower, Dec 13 2001; revised Apr 25 2006

Keywords

• nonn

Comments

Also number of aperiodic cyclic graphs with oriented edges on n nodes that can be turned over.

Examples

			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.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Transforms
• Index entries for sequences related to bracelets

Crossrefs

Cf. A053656.

Programs

PARI

\\ here b(n) is A053656.
b(n)={(sumdiv(n, d, eulerphi(d)*2^(n/d))/n + if(n%2==0, 2^(n/2-1)))/2}
a(n)={sumdiv(n, d, moebius(d)*b(n/d))} \\ Andrew Howroyd, Jun 14 2021

Formula

Moebius transform of A053656.

Extensions

Terms a(36) and beyond from Andrew Howroyd, Jun 14 2021