A323863 Number of n X n aperiodic binary arrays.

1, 2, 8, 486, 64800, 33554250, 68718675672, 562949953420302, 18446744060824780800, 2417851639229257812542976, 1267650600228226023797043513000, 2658455991569831745807614120560664598, 22300745198530623141521551172073990303938400

Offset

0,2

Comments

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Formula

a(n) = 2^(n^2) - (n+1)*2^n + 2*n if n is prime. - Robert Israel, Feb 04 2019

a(n) = n^2 * A323872(n). - Andrew Howroyd, Aug 21 2019

Example

The a(2) = 8 arrays are:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are (independently) aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

Mathematica

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}];
Table[Length[Select[(Partition[#, n]&)/@Tuples[{0, 1}, n^2], apermatQ]], {n, 4}]

Crossrefs

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

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

Sequence in context: A035129 A284603 A329685 * A003301 A000890 A033542

Adjacent sequences: A323860 A323861 A323862 * A323864 A323865 A323866

Keyword

nonn

Author

Gus Wiseman, Feb 04 2019

Extensions

a(5) from Robert Israel, Feb 04 2019

a(6)-a(7) from Giovanni Resta, Feb 05 2019

Terms a(8) and beyond from Andrew Howroyd, Aug 21 2019

Status

approved