A056290 Number of primitive (period n) n-bead necklaces with exactly five different colored beads.

0, 0, 0, 0, 24, 300, 2400, 15750, 92680, 510288, 2691600, 13793850, 69309240, 343499100, 1686135352, 8221421250, 39901776360, 193053923860, 932142850800, 4495236287850, 21664357532920, 104388118174500, 503044634004000, 2425003910574000, 11696087875731600

Offset

1,5

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

Formula

sum mu(d)*A056285(n/d) where d|n.

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, 5-j)*binomial(5, j)*(-1)^j, j=0..5):
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, 5 - j]*Binomial[5, j]*(-1)^j, {j, 0, 5}];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jun 06 2018, after Alois P. Heinz *)

Crossrefs

Cf. A001692.

Column k=5 of A254040.

Sequence in context: A162366 A153782 A073990 * A056285 A162686 A010976

Adjacent sequences: A056287 A056288 A056289 * A056291 A056292 A056293

Keyword

nonn

Author

Marks R. Nester

Status

approved