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 A059053 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. 10
 0, 0, 0, 0, 0, 0, 0, 1, 2, 7, 12, 31, 58, 126, 234, 484, 906, 1800, 3402, 6643, 12624, 24458, 46686, 90157, 172810, 333498, 641340, 1238671, 2388852, 4620006, 8932032, 17302033, 33522698, 65042526, 126258960, 245361172, 477091232 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Number of chiral pairs of set partitions of a cycle of n elements using exactly two different elements. - Robert A. Russell, Oct 02 2018 LINKS FORMULA a(n) = A000013(n) - A000011(n) = A000011(n) - A016116(n) = (A000013(n) - A016116(n))/2. From Robert A. Russell, Oct 02 2018: (Start) a(n) = (A056295(n)-A052551(n-2)) / 2 = A056295(n) - A056357(n) = A056357(n) - A052551(n-2). 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. G.f.: -x(1+2x)/(2-4x^2) - Sum_{d>0} phi(d) * log(1-2x^d) / (2d*(2-[2|d])). (End) EXAMPLE For a(7) = 1, the chiral pair is AAABABB-AAABBAB. For a(8) = 2 the chiral pairs are AAAABABB-AAAABBAB and AAABAABB-AAABBAAB. MATHEMATICA 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 *) CROSSREFS Cf. A056295 (oriented), A056357 (unoriented), A052551(n-2) (achiral). Sequence in context: A102371 A007230 A290234 * A032025 A088662 A073710 Adjacent sequences:  A059050 A059051 A059052 * A059054 A059055 A059056 KEYWORD nonn AUTHOR Henry Bottomley, Dec 21 2000 EXTENSIONS Name clarified by Robert A. Russell, Oct 02 2018 STATUS approved

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Last modified May 26 17:17 EDT 2019. Contains 323597 sequences. (Running on oeis4.)