OFFSET
0,4
COMMENTS
Number of n-bead necklaces using exactly two different colors. - Robert A. Russell, Sep 26 2018
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 788
Terry R. Payne and Luke Riley, From Necklaces to Coalitions: Fair and Self-Interested Distribution of Coalition Value Calculations, arXiv:2604.17057 [cs.GT], 2026. See p. 41.
Terry R. Payne, Luke Riley, Katie Atkinson, and Paul Dunne, Using Two Colour Necklaces to Fairly Allocate Coalition Value Calculations, 23rd IEEE/WIC Int'l Conf. Web Intel. Intel. Agent Tech. (WI-IAT 2024). See p. 7.
Shingo Saito, Tatsushi Tanaka, and Noriko Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values, J. Int. Seq. 14 (2011) # 11.2.4, Table 2.
FORMULA
G.f.: Sum_{j>=1} phi(j)/j*log(-(x^j-1)^2/(2*x^j-1)).
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=2 is the number of colors and S2 is the Stirling subset number A008277. - Robert A. Russell, Sep 26 2018
a(n) ~ 2^n / n. - Vaclav Kotesovec, Sep 25 2019
MAPLE
spec := [S, {B=Sequence(Z, 1 <= card), C=Prod(B, B), S= Cycle(C)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
k=2; Prepend[Table[k!DivisorSum[n, EulerPhi[#]StirlingS2[n/#, k]&]/n, {n, 1, 30}], 0] (* Robert A. Russell, Sep 26 2018 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
EXTENSIONS
More terms from Alois P. Heinz, Jan 25 2015
STATUS
approved