A032165 - OEIS

%I #32 May 19 2019 13:08:05

%S 10,45,330,2475,19998,166485,1428570,12498750,111111000,999989991,

%T 9090909090,83333249175,769230769230,7142856428565,66666666659934,

%U 624999993750000,5882352941176470,55555555499944500

%N Number of aperiodic necklaces of n beads of 10 colors.

%H Vincenzo Librandi, <a href="/A032165/b032165.txt">Table of n, a(n) for n = 1..500</a>

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>

%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]

%H <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>

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

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

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

%t 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 *)

%o (PARI) a(n) = sumdiv(n, d, moebius(d)*10^(n/d))/n; \\ _Andrew Howroyd_, Oct 13 2017

%Y Column 10 of A074650.

%K nonn

%O 1,1

%A _Christian G. Bower_