A052823 A simple grammar: cycles of pairs of sequences.

0, 0, 1, 2, 4, 6, 12, 18, 34, 58, 106, 186, 350, 630, 1180, 2190, 4114, 7710, 14600, 27594, 52486, 99878, 190744, 364722, 699250, 1342182, 2581426, 4971066, 9587578, 18512790, 35792566, 69273666, 134219794, 260301174, 505294126, 981706830, 1908881898

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

S. Saito, T. Tanaka, and N. 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

A000031 - 2.

Column k=2 of A087854.

Sequence in context: A007436 A052847 A331933 * A063516 A306315 A104352

Adjacent sequences: A052820 A052821 A052822 * A052824 A052825 A052826

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Extensions

More terms from Alois P. Heinz, Jan 25 2015

Status

approved