A032165 Number of aperiodic necklaces of n beads of 10 colors.

10, 45, 330, 2475, 19998, 166485, 1428570, 12498750, 111111000, 999989991, 9090909090, 83333249175, 769230769230, 7142856428565, 66666666659934, 624999993750000, 5882352941176470, 55555555499944500

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 10 of A074650.

Programs

Mathematica

f[d_]:=MoebiusMu[d] 10^(n/d)/n; a[n_]:=Total[f/@Divisors[n]]; a[0]=1; Table[a[n], {n, 1, 20}] (* Vincenzo Librandi, Oct 14 2017 *)

PARI

a(n) = sumdiv(n, d, moebius(d)*10^(n/d))/n; \\ Andrew Howroyd, Oct 13 2017

Formula

"CHK" (necklace, identity, unlabeled) transform of 10, 0, 0, 0...

a(n) = Sum_{d|n} mu(d)*10^(n/d)/n.

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 10*x^k))/k. - Ilya Gutkovskiy, May 19 2019