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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A320644 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 4 colors (subsets). 4
0, 0, 0, 0, 0, 2, 17, 84, 388, 1586, 6405, 24927, 96404, 368641, 1407515, 5357974, 20403120, 77699323, 296229485, 1130614092, 4321324766, 16539645539, 63397442097, 243352167691, 935420468092, 3600493932070, 13876442107403, 53546144395718, 206864753332164, 800067806813323, 3097590602034137, 12004772596768984, 46568647645538594, 180809553280920680 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.

Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.

There are nonrecursive formulas, generating functions, and computer programs for A056297 and A304974, which can be used in conjunction with the first formula.

LINKS

Table of n, a(n) for n=1..34.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

FORMULA

a(n) = (A056297(n) - A304974(n)) / 2 = A056297(n) - A056359(n) = A056359(n) - A304974(n).

a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=4 is number of colors or sets, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k)+Ach(n-2,k-1)+Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).

EXAMPLE

For a(6)=2, the chiral pairs are AABACD-AABCAD and AABCBD-AABCDC.

PROG

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]] (* A304972 *)

Adnk[d_, n_, k_] := Adnk[d, n, k] = If[n>0 && k>0, Adnk[d, n-1, k]k + DivisorSum[d, Adnk[d, n-1, k-#] &], Boole[n==0 && k==0]]

k=4; Table[DivisorSum[n, EulerPhi[#]Adnk[#, n/#, k]&]/(2n) - Ach[n, k]/2, {n, 40}]

CROSSREFS

Column 4 of A320747.

Cf. A056297 (oriented), A056359 (unoriented), A304974 (achiral).

Sequence in context: A079889 A053786 A181546 * A081744 A219757 A297727

Adjacent sequences:  A320641 A320642 A320643 * A320645 A320646 A320647

KEYWORD

nonn,easy

AUTHOR

Robert A. Russell, Oct 18 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 July 22 17:02 EDT 2019. Contains 325225 sequences. (Running on oeis4.)