login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=0..36.

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 *)

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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