A320643 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 3 colors (subsets).

0, 0, 0, 0, 0, 4, 12, 44, 137, 408, 1190, 3416, 9730, 27560, 78148, 221250, 627960, 1784038, 5081154, 14496956, 41455409, 118764600, 340919744, 980315700, 2823696150, 8145853520, 23533759241, 68081765650, 197206716570, 571906256808, 1660387879116, 4825525985408, 14037945170525, 40875277302720, 119122416494961, 347440682773324, 1014151818975190, 2962391932326680, 8659301777595196, 25328461701728194

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 A056296 and A304973, 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) = (A056296(n) - A304973(n)) / 2 = A056296(n) - A056358(n) = A056358(n) - A304973(n).

a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=3 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)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.

Mathematica

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=3; Table[DivisorSum[n, EulerPhi[#]Adnk[#, n/#, k]&]/(2n) - Ach[n, k]/2, {n, 40}]

Crossrefs

Column 3 of A320647.

Cf. A056296 (oriented), A056358 (unoriented), A304973 (achiral).

Sequence in context: A149359 A259223 A167402 * A060897 A005190 A149360

Adjacent sequences: A320640 A320641 A320642 * A320644 A320645 A320646

Keyword

nonn,easy

Author

Robert A. Russell, Oct 18 2018

Status

approved