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.

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

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

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Illustration of initial terms

Index entries for sequences related to necklaces

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 *)

Prog

(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

Crossrefs

Column 2 of A320647 and A320742.

Cf. A056295 (oriented), A056357 (unoriented), A052551(n-2) (achiral).

Sequence in context: A290234 A327734 A308706 * 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