A032295 Number of aperiodic bracelets (turn over necklaces) with n beads of 4 colors.

4, 6, 16, 45, 132, 404, 1296, 4380, 15064, 53622, 192696, 703895, 2589300, 9606744, 35824088, 134297280, 505421340, 1909194056, 7234153416, 27489073899, 104717489748, 399827555604, 1529763696816

Offset

1,1

Links

Table of n, a(n) for n=1..23.

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

Formula

MOEBIUS transform of A032275.

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)

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, 4], {x, 0, mx}], x] (* Herbert Kociemba, Nov 28 2016 *)

Crossrefs

Column 4 of A276550.

Sequence in context: A165799 A231998 A056421 * A072279 A038236 A223269

Adjacent sequences: A032292 A032293 A032294 * A032296 A032297 A032298

Keyword

nonn

Author

Christian G. Bower

Status

approved