A056283 Number of n-bead necklaces with exactly three different colored beads.

0, 0, 2, 9, 30, 91, 258, 729, 2018, 5613, 15546, 43315, 120750, 338259, 950062, 2678499, 7573350, 21480739, 61088874, 174184755, 497812638, 1425847623, 4092087522, 11765822365, 33887517870, 97756387365, 282414624746, 816999710223, 2366509198350, 6862930841141

Offset

1,3

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) = A001867(n) - 3*A000031(n) + 3.

From Robert A. Russell, Sep 26 2018: (Start)

a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=3 is the number of colors and S2 is the Stirling subset number A008277.

G.f.: -Sum_{d>0} (phi(d)/d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=3 is the number of colors. (End)

Example

For n=3, the two necklaces are ABC and ACB.

Mathematica

k=3; Table[k!DivisorSum[n, EulerPhi[#]StirlingS2[n/#, k]&]/n, {n, 1, 30}] (* Robert A. Russell, Sep 26 2018 *)

Crossrefs

Cf. A000031, A001867, A052823.

Column k=3 of A087854.

Sequence in context: A056288 A261174 A273652 * A192518 A277241 A201164

Adjacent sequences: A056280 A056281 A056282 * A056284 A056285 A056286

Keyword

nonn

Author

Marks R. Nester

Status

approved