# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a323872 Showing 1-1 of 1 %I A323872 #10 Aug 22 2019 22:14:39 %S A323872 1,2,2,54,4050,1342170,1908852102,11488774559598,288230375950387200, %T A323872 29850020237398244599296,12676506002282260237970435130, %U A323872 21970710674130840874443091905460038,154866286100907105149455216472736043777350,4427744605404865645682169434028029029963535277450 %N A323872 Number of n X n aperiodic binary toroidal necklaces. %C A323872 The 1-dimensional (Lyndon word) case is A001037. %C A323872 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 A323872 Andrew Howroyd, Table of n, a(n) for n = 0..50 %e A323872 Inequivalent representatives of the a(2) = 2 aperiodic necklaces: %e A323872 [0 0] [0 1] %e A323872 [0 1] [1 1] %e A323872 Inequivalent representatives of the a(3) = 54 aperiodic necklaces: %e A323872 000 000 000 000 000 000 000 000 000 %e A323872 000 000 001 001 001 001 001 001 001 %e A323872 001 011 001 010 011 100 101 110 111 %e A323872 . %e A323872 000 000 000 000 000 000 000 000 000 %e A323872 011 011 011 011 011 011 011 111 111 %e A323872 001 010 011 100 101 110 111 001 011 %e A323872 . %e A323872 001 001 001 001 001 001 001 001 001 %e A323872 001 001 001 001 001 001 010 010 010 %e A323872 010 011 100 101 110 111 011 101 110 %e A323872 . %e A323872 001 001 001 001 001 001 001 001 001 %e A323872 010 011 011 011 011 011 100 100 100 %e A323872 111 010 011 101 110 111 011 110 111 %e A323872 . %e A323872 001 001 001 001 001 001 001 001 001 %e A323872 101 101 101 101 110 110 110 110 111 %e A323872 011 101 110 111 011 101 110 111 011 %e A323872 . %e A323872 001 001 001 011 011 011 011 011 011 %e A323872 111 111 111 011 011 011 101 110 111 %e A323872 101 110 111 101 110 111 111 111 111 %t A323872 apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]; %t A323872 neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]]; %t A323872 Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],And[apermatQ[#],neckmatQ[#]]&]],{n,4}] %Y A323872 Main diagonal of A323861. %Y A323872 Cf. A000031, A000740, A001037, A027375, A059966, A179043, A184271, A323351. %Y A323872 Cf. A323859, A323860, A323865, A323866, A323871. %K A323872 nonn %O A323872 0,2 %A A323872 _Gus Wiseman_, Feb 04 2019 %E A323872 Terms a(5) and beyond from _Andrew Howroyd_, Aug 21 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE