A032166 Number of aperiodic necklaces of n beads of 11 colors.

11, 55, 440, 3630, 32208, 295020, 2783880, 26793030, 261994040, 2593726344, 25937424600, 261535549220, 2655593241840, 27124986721140, 278483211283552, 2871858103075830, 29732178147017280

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Comments

Number of monic irreducible polynomials of degree n over GF(11). # Robert Israel, Jan 07 2015

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 11 of A074650.

Programs

Maple

f:= (n,p) -> add(numtheory:-mobius(d)*p^(n/d),d=numtheory:-divisors(n))/n:
seq(f(n,11), n=1..100); # Robert Israel, Jan 07 2015

Mathematica

f[d_]:=MoebiusMu[d] 11^(n/d)/n; a[n_]:=Total[f/@Divisors[n]]; a[0]=1; Table[a[n], {n, 1, 30}] (* Vincenzo Librandi, Oct 14 2017 *)

PARI

a(n) = sumdiv(n, d, moebius(d)*11^(n/d))/n; \\ Michel Marcus, Jan 07 2015

Formula

"CHK" (necklace, identity, unlabeled) transform of 11, 0, 0, 0...

a(n) = Sum_{d|n} mu(d)*11^(n/d)/n.

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 11*x^k))/k. - Ilya Gutkovskiy, May 19 2019