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

1,4

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

N. J. A. Sloane, Transforms

Index entries for sequences related to bracelets

Formula

Moebius transform of A053656.

Example

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.

Prog

(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

Crossrefs

Cf. A053656.

Sequence in context: A182522 A165647 A191398 * A224958 A304912 A018499

Adjacent sequences: A066310 A066311 A066312 * A066314 A066315 A066316

Keyword

nonn

Author

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

Extensions

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

Status

approved