The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A320743 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets). 4
 0, 0, 0, 0, 0, 4, 13, 46, 144, 420, 1221, 3474, 9856, 27794, 78632, 222156, 629760, 1787440, 5087797, 14509580, 41479867, 118811286, 341009901, 980488510, 2824029648, 8146494860, 23534997912, 68084154502, 197211336576, 571915188840, 1660405181149, 4825559508106, 14038010213051, 40875403561680, 119122661856133, 347441159864556, 1014152747485696 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Two color patterns are equivalent if the colors are permuted. 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 A002076 and A182522, which can be used in conjunction with the first formula. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. FORMULA a(n) = (A002076(n) - A182522(n)) / 2 = A002076(n) - A056353(n) = A056353(n) - A182522(n). a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=3 is the maximum number of colors, 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)). a(n) = A059053(n) + A320643(n). EXAMPLE For a(6)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC. MATHEMATICA 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]] 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 *) k=3; Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#, n/#, j]&]/n - Ach[n, j])/2, {j, k}], {n, 40}] CROSSREFS Column 3 of A320742. Cf. A002076 (oriented), A056353 (unoriented), A182522 (achiral). Sequence in context: A155328 A096353 A034553 * A104460 A095128 A149433 Adjacent sequences:  A320740 A320741 A320742 * A320744 A320745 A320746 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 21 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.

Last modified July 24 11:06 EDT 2021. Contains 346273 sequences. (Running on oeis4.)