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A052823 A simple grammar: cycles of pairs of sequences. 11
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 (list; graph; refs; listen; history; text; internal format)
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, 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(numtheory[phi](j[1])/j[1]*log(-(x^j[1]-1)^2/(2*x^j[1]-1)), j[1]=1 .. infinity).

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

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: A259941 A007436 A052847 * A063516 A104352 A133488

Adjacent sequences:  A052820 A052821 A052822 * A052824 A052825 A052826

KEYWORD

easy,nonn,changed

AUTHOR

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

EXTENSIONS

More terms from Alois P. Heinz, Jan 25 2015

STATUS

approved

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Last modified October 18 09:28 EDT 2018. Contains 316317 sequences. (Running on oeis4.)