A032294 Number of aperiodic bracelets (turnover necklaces) with n beads of 3 colors.

3, 3, 7, 15, 36, 79, 195, 477, 1209, 3168, 8415, 22806, 62412, 172887, 481552, 1351485, 3808080, 10780653, 30615351, 87226932, 249144506, 713378655, 2046856563, 5884468110, 16946569332, 48883597728, 141217159239

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Links

• C. G. Bower, Transforms (2)
• 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]
• N. J. A. Sloane, Transforms
• Index entries for sequences related to bracelets

Crossrefs

Column 3 of A276550.

Programs

Mathematica

mx=40;gf[x_,k_]:=Sum[ MoebiusMu[n]*(-Log[1-k*x^n]/n+Sum[Binomial[k,i]x^(n i),{i,0,2}]/( 1-k x^(2n)))/2,{n,mx}]; CoefficientList[Series[gf[x,3],{x,0,mx}],x] (* Herbert Kociemba, Nov 28 2016 *)

PARI

a(x, k) = sum(n=1, 40, moebius(n) * (-log(1 - k*x^n )/n + sum(i=0, 2, binomial(k, i) * x^(n*i)) / (1 - k* x^(2*n)))/2);
Vec(a(x, 3) + O(x^41)) \\ Indranil Ghosh, Mar 29 2017

Formula

MOEBIUS transform of A027671.

From Herbert Kociemba, Nov 28 2016: (Start)

More generally, gf(k) is the g.f. for the number of bracelets with primitive period n and beads of k colors.

gf(k): Sum_{n>=1} mu(n)*( -log(1-k*x^n)/n + Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)) )/2. (End)