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 A324513 Number of aperiodic cycle necklaces with n vertices. 8
 1, 0, 0, 0, 2, 7, 51, 300, 2238, 18028, 164945, 1662067, 18423138, 222380433, 2905942904, 40864642560, 615376173176, 9880203467184, 168483518571789, 3041127459127222, 57926238289894992, 1161157775616335125, 24434798429947993043, 538583682037962702384 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000740, A000939, A001037 (binary Lyndon words), A008965, A059966 (Lyndon compositions), A060223 (normal Lyndon words), A061417, A064852 (if cycle is oriented), A086675, A192332, A275527, A323866 (aperiodic toroidal arrays), A323871. Cf. A306669, A324461, A324462, A324512, A324514. Sequence in context: A186860 A139008 A058721 * A086902 A265042 A249754 Adjacent sequences:  A324510 A324511 A324512 * A324514 A324515 A324516 KEYWORD nonn AUTHOR Gus Wiseman, Mar 04 2019 EXTENSIONS Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019 STATUS approved

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Last modified October 19 03:34 EDT 2019. Contains 328211 sequences. (Running on oeis4.)