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%I #10 Aug 23 2019 00:58:41
%S 1,2,8,53,216,3112,13512,272844,2362412,40898808,295024104,
%T 14045779864,81055130520,3040383692328,61408850927280,
%U 1661142087743940,15337737297545400,1128511554416582908,9768588138876674856,803306338873264137240,15452347618762680730384
%N Number of aperiodic toroidal necklaces of size n whose entries cover an initial interval of positive integers.
%C The 1-dimensional (Lyndon word) case is A060223.
%C We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.
%H Andrew Howroyd, <a href="/A323871/b323871.txt">Table of n, a(n) for n = 1..200</a>
%H S. N. Ethier, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ethier/ethier2.html">Counting toroidal binary arrays</a>, J. Int. Seq. 16 (2013) #13.4.7.
%e The a(3) = 8 aperiodic toroidal necklaces:
%e [1 2 3] [1 3 2] [1 2 2] [1 1 2]
%e .
%e [1] [1] [1] [1]
%e [2] [3] [2] [1]
%e [3] [2] [2] [2]
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
%t apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
%t neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
%t Table[Length[Select[nrmmats[n],neckmatQ[#]&&apermatQ[#]&]],{n,6}]
%o (GAP) List([1..30], A323871); # See A323861 for code; _Andrew Howroyd_, Aug 21 2019
%Y Cf. A000670, A000740, A008965, A060223.
%Y Cf. A323858, A323859, A323860, A323861, A323866, A323867, A323868, A323870.
%K nonn
%O 1,2
%A _Gus Wiseman_, Feb 04 2019
%E Terms a(9) and beyond from _Andrew Howroyd_, Aug 21 2019
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