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A320643
Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 3 colors (subsets).
6
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
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
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 18 2018
STATUS
approved