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%I #30 Nov 04 2019 02:25:55
%S 0,0,0,0,0,0,0,1,2,7,12,31,58,126,234,484,906,1800,3402,6643,12624,
%T 24458,46686,90157,172810,333498,641340,1238671,2388852,4620006,
%U 8932032,17302033,33522698,65042526,126258960,245361172,477091232
%N Number of chiral pairs of necklaces with n beads and two colors (color complements being equivalent); i.e., turning the necklace over neither leaves it unchanged nor simply swaps the colors.
%C Number of chiral pairs of set partitions of a cycle of n elements using exactly two different elements. - _Robert A. Russell_, Oct 02 2018
%H Andrew Howroyd, <a href="/A059053/b059053.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/A059053/a059053.gif">Illustration of initial terms</a>
%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%F a(n) = A000013(n) - A000011(n) = A000011(n) - A016116(n) = (A000013(n) - A016116(n))/2.
%F From _Robert A. Russell_, Oct 02 2018: (Start)
%F a(n) = (A056295(n)-A052551(n-2)) / 2 = A056295(n) - A056357(n) = A056357(n) - A052551(n-2).
%F a(n) = -S2(1+floor(n/2),2) + (1/2n) * Sum_{d|n} phi(d) * S2(n/d+[2|d],2), where S2 is a Stirling subset number A008277.
%F G.f.: -x(1+2x)/(2-4x^2) - Sum_{d>0} phi(d) * log(1-2x^d) / (2d*(2-[2|d])).
%F (End)
%e For a(7) = 1, the chiral pair is AAABABB-AAABBAB.
%e For a(8) = 2, the chiral pairs are AAAABABB-AAAABBAB and AAABAABB-AAABBAAB.
%t Prepend[Table[DivisorSum[n, EulerPhi[#] StirlingS2[n/# + If[Divisible[#,2],1,0], 2] &] / (2n) - StirlingS2[1+Floor[n/2],2] / 2, {n, 1, 40}],0] (* _Robert A. Russell_, Oct 02 2018 *)
%o (PARI) a(n) = {if(n<1, 0, (sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (2*n) - 2^(n\2))/2)}; \\ _Andrew Howroyd_, Nov 03 2019
%Y Column 2 of A320647 and A320742.
%Y Cf. A056295 (oriented), A056357 (unoriented), A052551(n-2) (achiral).
%K nonn
%O 0,9
%A _Henry Bottomley_, Dec 21 2000
%E Name clarified by _Robert A. Russell_, Oct 02 2018
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