%I #10 Aug 22 2019 22:14:39

%S 1,2,2,54,4050,1342170,1908852102,11488774559598,288230375950387200,

%T 29850020237398244599296,12676506002282260237970435130,

%U 21970710674130840874443091905460038,154866286100907105149455216472736043777350,4427744605404865645682169434028029029963535277450

%N Number of n X n aperiodic binary toroidal necklaces.

%C The 1-dimensional (Lyndon word) case is A001037.

%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="/A323872/b323872.txt">Table of n, a(n) for n = 0..50</a>

%e Inequivalent representatives of the a(2) = 2 aperiodic necklaces:

%e [0 0] [0 1]

%e [0 1] [1 1]

%e Inequivalent representatives of the a(3) = 54 aperiodic necklaces:

%e 000 000 000 000 000 000 000 000 000

%e 000 000 001 001 001 001 001 001 001

%e 001 011 001 010 011 100 101 110 111

%e .

%e 000 000 000 000 000 000 000 000 000

%e 011 011 011 011 011 011 011 111 111

%e 001 010 011 100 101 110 111 001 011

%e .

%e 001 001 001 001 001 001 001 001 001

%e 001 001 001 001 001 001 010 010 010

%e 010 011 100 101 110 111 011 101 110

%e .

%e 001 001 001 001 001 001 001 001 001

%e 010 011 011 011 011 011 100 100 100

%e 111 010 011 101 110 111 011 110 111

%e .

%e 001 001 001 001 001 001 001 001 001

%e 101 101 101 101 110 110 110 110 111

%e 011 101 110 111 011 101 110 111 011

%e .

%e 001 001 001 011 011 011 011 011 011

%e 111 111 111 011 011 011 101 110 111

%e 101 110 111 101 110 111 111 111 111

%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[(Partition[#,n]&)/@Tuples[{0,1},n^2],And[apermatQ[#],neckmatQ[#]]&]],{n,4}]

%Y Main diagonal of A323861.

%Y Cf. A000031, A000740, A001037, A027375, A059966, A179043, A184271, A323351.

%Y Cf. A323859, A323860, A323865, A323866, A323871.

%K nonn

%O 0,2

%A _Gus Wiseman_, Feb 04 2019

%E Terms a(5) and beyond from _Andrew Howroyd_, Aug 21 2019