A032167 Number of aperiodic necklaces of n beads of 12 colors.

12, 66, 572, 5148, 49764, 497354, 5118828, 53745120, 573308736, 6191711526, 67546215516, 743008120140, 8230246567620, 91708459194066, 1027134771622388, 11555266154065920, 130506535690613940

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..500
• C. G. Bower, Transforms (2)
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
• Index entries for sequences related to Lyndon words

Crossrefs

Column 12 of A074650.

Programs

Mathematica

f[d_]:=MoebiusMu[d] 12^(n/d)/n; a[n_]:=Total[f/@Divisors[n]]; a[0]=1; Table[a[n], {n, 1, 30}] (* Vincenzo Librandi, Oct 14 2017 *)

Formula

"CHK" (necklace, identity, unlabeled) transform of 12, 0, 0, 0...

a(n) = Sum_{d|n} mu(d)*12^(n/d)/n.

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 12*x^k))/k. - Ilya Gutkovskiy, May 19 2019