OFFSET
1,8
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.
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) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=5 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(8)=9, the chiral pairs are AABACDEF-AABCDEAF, AABCADEF-AABCDAEF, AABCBDEF-AABCDEFE, AABCDBEF-AABCDEFD, AABCDEBF-AABCDEFC, AABCDCEF-AABCDEDF, ABACDEBF-ABACDEBF, ABCADBEF-ABCADECF, and ABCDAEBF-ABCADBEF.
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=6; Table[DivisorSum[n, EulerPhi[#]Adnk[#, n/#, k]&]/(2n) - Ach[n, k]/2, {n, 40}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 19 2018
STATUS
approved