A056289 Number of primitive (period n) n-bead necklaces with exactly four different colored beads.

0, 0, 0, 6, 48, 260, 1200, 5100, 20720, 81828, 318000, 1222870, 4675440, 17813820, 67769504, 257695800, 980240880, 3731732200, 14222737200, 54278498154, 207438936800, 793940157900, 3043140078000, 11681056021300, 44900438149248, 172824327151140, 666070256468960

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

Turning over the necklace is not allowed.

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

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A027377.

Column k=4 of A254040.

Programs

Maple

with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(mobius(n/d)*k^d, d=divisors(n))/n)
    end:
a:= n-> add(b(n, 4-j)*binomial(4, j)*(-1)^j, j=0..4):
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 25 2015

Mathematica

b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^#&]/n]; a[n_] := Sum[b[n, 4-j]*Binomial[4, j]*(-1)^j, {j, 0, 4}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 20 2017, after Alois P. Heinz *)

Formula

a(n) = Sum_{d|n} mu(d)*A056284(n/d).