A056284 Number of n-bead necklaces with exactly four different colored beads.

0, 0, 0, 6, 48, 260, 1200, 5106, 20720, 81876, 318000, 1223136, 4675440, 17815020, 67769552, 257700906, 980240880, 3731753180, 14222737200, 54278580036, 207438938000, 793940475900, 3043140078000, 11681057249536, 44900438149296, 172824331826580, 666070256489680

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) = A001868(n) - 4*A001867(n) + 6*A000031(n) - 4.

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

a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=4 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=4 is the number of colors. (End)

Example

For n=4, the six necklaces are ABCD, ABDC, ACBD, ACDB, ADBC and ADCB.

Mathematica

k=4; Table[k!DivisorSum[n, EulerPhi[#]StirlingS2[n/#, k]&]/n, {n, 1, 30}] (* Robert A. Russell, Sep 26 2018 *)

Prog

(PARI) a(n) = my(k=4); (k!/n)*sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2)); \\ Michel Marcus, Sep 27 2018

Crossrefs

Cf. A001868.

Column k=4 of A087854.

Sequence in context: A262354 A052771 A056289 * A366622 A293967 A246587

Adjacent sequences: A056281 A056282 A056283 * A056285 A056286 A056287

Keyword

nonn

Author

Marks R. Nester

Status

approved