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Number of n X n aperiodic binary arrays.
7

%I #15 Aug 22 2019 20:47:36

%S 1,2,8,486,64800,33554250,68718675672,562949953420302,

%T 18446744060824780800,2417851639229257812542976,

%U 1267650600228226023797043513000,2658455991569831745807614120560664598,22300745198530623141521551172073990303938400

%N Number of n X n aperiodic binary arrays.

%C 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="/A323863/b323863.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) = 2^(n^2) - (n+1)*2^n + 2*n if n is prime. - _Robert Israel_, Feb 04 2019

%F a(n) = n^2 * A323872(n). - _Andrew Howroyd_, Aug 21 2019

%e The a(2) = 8 arrays are:

%e [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]

%e [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]

%e Note that the following are not aperiodic even though their row and column sequences are (independently) aperiodic:

%e [1 0] [0 1]

%e [0 1] [1 0]

%t apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];

%t Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],apermatQ]],{n,4}]

%Y Cf. A000031, A000740, A027375, A179043, A265627, A323351.

%Y Cf. A323860, A323862, A323864, A323865, A323867, A323869.

%K nonn

%O 0,2

%A _Gus Wiseman_, Feb 04 2019

%E a(5) from _Robert Israel_, Feb 04 2019

%E a(6)-a(7) from _Giovanni Resta_, Feb 05 2019

%E Terms a(8) and beyond from _Andrew Howroyd_, Aug 21 2019