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%I #39 Sep 12 2019 21:11:56
%S 2,1,0,0,0,1,2,6,14,30,62,127,252,493,968,1860,3600,6902,13286,25446,
%T 48914,93775,180314,346420,666996,1284318,2477328,4781007,9240012,
%U 17870709,34604066,67058880,130084990,252545160,490722342
%N Number of identity bracelets of n beads of 2 colors.
%C For n > 2, a(n) is also number of asymmetric bracelets with n beads of two colors. - _Herbert Kociemba_, Nov 29 2016
%H Andrew Howroyd, <a href="/A032239/b032239.txt">Table of n, a(n) for n = 1..1000</a>
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H Petros Hadjicostas, <a href="/A032239/a032239.pdf">Formulas for chiral bracelets</a>, 2019; see Section 5.
%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F "DHK" (bracelet, identity, unlabeled) transform of 2, 0, 0, 0...
%F From _Herbert Kociemba_, Nov 29 2016: (Start)
%F More generally, gf(k) is the g.f. for the number of asymmetric bracelets with n beads of k colors.
%F gf(k): Sum_{n>=1} mu(n) * ( -log(1 - k*x^n)/n - Sum_{i=0..2} binomial(k,i) * x^(n*i)/(1 - k*x^(2*n)) )/2. (End)
%t m = 2; (* asymmetric bracelets of n beads of m colors *) Table[Sum[MoebiusMu[d] (m^(n/d)/n - If[OddQ[n/d], m^((n/d + 1)/2), ((m + 1) m^(n/(2 d))/2)]), {d, Divisors[n]}]/2, {n, 3, 20}] (* _Robert A. Russell_, Mar 18 2013 *)
%t mx=40;gf[x_,k_]:=Sum[MoebiusMu[n]*(-Log[1-k*x^n]/n-Sum[Binomial[k,i]x^(n i),{i,0,2}]/(1-k x^(2n)))/2,{n,mx}];ReplacePart[Rest[CoefficientList[Series[gf[x,2],{x,0,mx}],x]],{1->2,2->1}] (* _Herbert Kociemba_, Nov 29 2016 *)
%o (PARI) a(n)={if(n<3, binomial(2,n), sumdiv(n, d, moebius(n/d)*(2^d/n - if(d%2, 2^((d+1)/2), 3*2^(d/2)/2)))/2)} \\ _Andrew Howroyd_, Sep 12 2019
%Y Column k=2 of A309528 and A309651 for n >= 3.
%Y Row sums of A308583 for n >= 3.
%Y Cf. A032337, A203399.
%K nonn
%O 1,1
%A _Christian G. Bower_
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