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A056289 Number of primitive (period n) n-bead necklaces with exactly four different colored beads. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

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

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-Fran├žois Alcover, Feb 20 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A027377.

Column k=4 of A254040.

Sequence in context: A260344 A262354 A052771 * A056284 A246587 A229504

Adjacent sequences:  A056286 A056287 A056288 * A056290 A056291 A056292

KEYWORD

nonn

AUTHOR

Marks R. Nester (nesterm(AT)dpi.qld.gov.au)

STATUS

approved

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Last modified June 22 23:15 EDT 2017. Contains 288633 sequences.