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A052823
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A simple grammar: cycles of pairs of sequences.
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11
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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
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OFFSET
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0,4
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COMMENTS
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Number of n-bead necklaces using exactly two different colors. - Robert A. Russell, Sep 26 2018
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LINKS
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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.
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FORMULA
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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
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MAPLE
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spec := [S, {B=Sequence(Z, 1 <= card), C=Prod(B, B), S= Cycle(C)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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k=2; Prepend[Table[k!DivisorSum[n, EulerPhi[#]StirlingS2[n/#, k]&]/n, {n, 1, 30}], 0] (* Robert A. Russell, Sep 26 2018 *)
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CROSSREFS
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A000031 - 2.
Column k=2 of A087854.
Sequence in context: A007436 A052847 A331933 * A063516 A306315 A104352
Adjacent sequences: A052820 A052821 A052822 * A052824 A052825 A052826
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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More terms from Alois P. Heinz, Jan 25 2015
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STATUS
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approved
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