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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 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

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Last modified August 8 08:27 EDT 2020. Contains 336293 sequences. (Running on oeis4.)