A032239 Number of identity bracelets of n beads of 2 colors.

2, 1, 0, 0, 0, 1, 2, 6, 14, 30, 62, 127, 252, 493, 968, 1860, 3600, 6902, 13286, 25446, 48914, 93775, 180314, 346420, 666996, 1284318, 2477328, 4781007, 9240012, 17870709, 34604066, 67058880, 130084990, 252545160, 490722342

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Comments

For n > 2, a(n) is also number of asymmetric bracelets with n beads of two colors. - Herbert Kociemba, Nov 29 2016

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• C. G. Bower, Transforms (2)
• Petros Hadjicostas, Formulas for chiral bracelets, 2019; see Section 5.
• 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 bracelets

Crossrefs

Column k=2 of A309528 and A309651 for n >= 3.

Row sums of A308583 for n >= 3.

Cf. A032337, A203399.

Programs

Mathematica

m = 2; (* asymmetric bracelets of n beads of m colors *) Table[Sum[MoebiusMu[d] (m^(n/d)/n - If[OddQ[n/d], m^((n/d + 1)/2), ((m + 1) m^(n/(2 d))/2)]), {d, Divisors[n]}]/2, {n, 3, 20}] (* Robert A. Russell, Mar 18 2013 *)
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}];ReplacePart[Rest[CoefficientList[Series[gf[x,2],{x,0,mx}],x]],{1->2,2->1}] (* Herbert Kociemba, Nov 29 2016 *)

PARI

a(n)={if(n<3, binomial(2,n), sumdiv(n, d, moebius(n/d)*(2^d/n - if(d%2, 2^((d+1)/2), 3*2^(d/2)/2)))/2)} \\ Andrew Howroyd, Sep 12 2019

Formula

"DHK" (bracelet, identity, unlabeled) transform of 2, 0, 0, 0...

From Herbert Kociemba, Nov 29 2016: (Start)

More generally, gf(k) is the g.f. for the number of asymmetric bracelets with n 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)