A056347 Number of primitive (period n) bracelets using a maximum of six different colored beads.

6, 15, 50, 210, 882, 4220, 20640, 107100, 563730, 3036411, 16514100, 90778485, 502474350, 2799199380, 15673672238, 88162569180, 497847963690, 2821127257950, 16035812864940, 91404065292036

Offset

1,1

Comments

Turning over will not create a new bracelet.

References

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Links

Table of n, a(n) for n=1..20.

Formula

sum mu(d)*A056341(n/d) where d|n.

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

Crossrefs

Column 6 of A276550.

Cf. A032164.

Sequence in context: A318414 A106272 A056423 * A271332 A244024 A122365

Adjacent sequences: A056344 A056345 A056346 * A056348 A056349 A056350

Keyword

nonn

Author

Marks R. Nester

Status

approved