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, 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
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from Alois P. Heinz, Jan 25 2015
STATUS
approved