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

Author

Marks R. Nester

Keywords

• nonn

Comments

Turning over the necklace is not allowed.

Examples

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

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. A000031, A001867, A052823.

Column k=3 of A087854.

Programs

Mathematica

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

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)