A203399 -id:A203399 - cp's OEIS frontend

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

Christian G. Bower

nonn

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

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

Column k=2 of A309528 and A309651 for n >= 3.
Row sums of A308583 for n >= 3.
Cf. A032337, A203399.

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 *)

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

"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)

A203398 T(n,k), a triangular array read by rows, is the number of classes of equivalent 2-color n-bead necklaces (turning over is not allowed) that have k necklaces.

Original entry on oeis.org

2, 2, 1, 2, 0, 2, 2, 1, 0, 3, 2, 0, 0, 0, 6, 2, 1, 2, 0, 0, 9, 2, 0, 0, 0, 0, 0, 18, 2, 1, 0, 3, 0, 0, 0, 30, 2, 0, 2, 0, 0, 0, 0, 0, 56, 2, 1, 0, 0, 6, 0, 0, 0, 0, 99, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 186, 2, 1, 2, 3, 0, 9, 0, 0, 0, 0, 0, 335

Offset: 1

Geoffrey Critzer, Jan 01 2012

Equivalently, the cyclic group of order n acts on the set of length n binary sequences. T(n,k) is the number of orbits that have k elements.

			  2
  2  1
  2  0  2
  2  1  0  3
  2  0  0  0  6
  2  1  2  0  0  9
  2  0  0  0  0  0  18
  2  1  0  3  0  0  0  30
  2  0  2  0  0  0  0  0  56
  2  1  0  0  6  0  0  0  0  99
  2  0  0  0  0  0  0  0  0  0  186
  2  1  2  3  0  9  0  0  0  0  0   335

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Combinatorial Generation Algorithm Algorithm 4.24, p. 95.

A000031 (row sums), T(n,n) = A001037, T(n,n) = A064535 when n is prime, T(n,k) = A001037(k) when k divides n.

Cf. A203399.

Needs["Combinatorica`"];
f[list_] := Sort[NestList[RotateLeft, list, Length[list]-1]]; Flatten[Table[Distribution[Map[Length, Map[Union, Union[Map[f, Strings[{0, 1}, n]]]]], Range[n]], {n, 1, 12}]]

cp's OEIS Frontend

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

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