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%I #18 Nov 04 2019 02:17:51
%S 0,0,0,0,0,4,12,44,137,408,1190,3416,9730,27560,78148,221250,627960,
%T 1784038,5081154,14496956,41455409,118764600,340919744,980315700,
%U 2823696150,8145853520,23533759241,68081765650,197206716570,571906256808,1660387879116,4825525985408,14037945170525,40875277302720,119122416494961,347440682773324,1014151818975190,2962391932326680,8659301777595196,25328461701728194
%N Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 3 colors (subsets).
%C Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
%C Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
%C There are nonrecursive formulas, generating functions, and computer programs for A056296 and A304973, which can be used in conjunction with the first formula.
%H Andrew Howroyd, <a href="/A320643/b320643.txt">Table of n, a(n) for n = 1..200</a>
%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%F a(n) = (A056296(n) - A304973(n)) / 2 = A056296(n) - A056358(n) = A056358(n) - A304973(n).
%F 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)).
%e For a(6)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.
%t 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 *)
%t 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]]
%t k=3; Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n) - Ach[n,k]/2,{n,40}]
%Y Column 3 of A320647.
%Y Cf. A056296 (oriented), A056358 (unoriented), A304973 (achiral).
%K nonn,easy
%O 1,6
%A _Robert A. Russell_, Oct 18 2018
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