OFFSET
1,5
COMMENTS
We define an aperiodic cycle necklace to be an equivalence class of (labeled, undirected) Hamiltonian cycles under rotation of the vertices such that all n of these rotations are distinct.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
FORMULA
a(n) = A324512(n)/n.
a(2*n+1) = A064852(2*n+1)/2 for n > 0; a(2*n) = (A064852(2*n) - A002866(n-1))/2 for n > 1. - Andrew Howroyd, Aug 16 2019
MATHEMATICA
rotgra[g_, m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m, 1, k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#, 2, 1, 1]]&/@Permutations[Range[n]]], #==First[Sort[Table[Nest[rotgra[#, n]&, #, j], {j, n}]]]&&UnsameQ@@Table[Nest[rotgra[#, n]&, #, j], {j, n}]&]], {n, 8}]
PROG
(PARI) a(n)={if(n<3, n==0||n==1, (if(n%2, 0, -(n/2-1)!*2^(n/2-2)) + sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2))/2)} \\ Andrew Howroyd, Aug 19 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 04 2019
EXTENSIONS
Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019
STATUS
approved