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%I #36 Sep 16 2020 10:37:01
%S 0,0,1,2,4,6,12,18,34,58,106,186,350,630,1180,2190,4114,7710,14600,
%T 27594,52486,99878,190744,364722,699250,1342182,2581426,4971066,
%U 9587578,18512790,35792566,69273666,134219794,260301174,505294126,981706830,1908881898
%N A simple grammar: cycles of pairs of sequences.
%C Number of n-bead necklaces using exactly two different colors. - _Robert A. Russell_, Sep 26 2018
%H Alois P. Heinz, <a href="/A052823/b052823.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=788">Encyclopedia of Combinatorial Structures 788</a>
%H S. Saito, T. Tanaka, and N. Wakabayashi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Saito/saito22.html">Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values </a>, J. Int. Seq. 14 (2011) # 11.2.4, Table 2.
%F G.f.: Sum_{j>=1} phi(j)/j*log(-(x^j-1)^2/(2*x^j-1)).
%F 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
%F a(n) ~ 2^n / n. - _Vaclav Kotesovec_, Sep 25 2019
%p spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= Cycle(C)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
%t k=2; Prepend[Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}],0] (* _Robert A. Russell_, Sep 26 2018 *)
%Y A000031 - 2.
%Y Column k=2 of A087854.
%K easy,nonn
%O 0,4
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _Alois P. Heinz_, Jan 25 2015
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