A323864 Number of aperiodic binary arrays of size n.

1, 2, 4, 12, 32, 60, 216, 252, 912, 1494, 3960, 4092, 23904, 16380, 65016, 130920, 324960, 262140, 1569132, 1048572, 6281280, 8388072, 16769016, 16777212, 134150880, 100663050, 268402680, 536865840, 1610449344, 1073741820, 8589664080, 4294967292, 25768888320

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..200

Formula

a(n) = Sum_{d|n} A323860(d, n/d). - Andrew Howroyd, Aug 21 2019

Example

The a(4) = 32 arrays:
  [0001][0010][0011][0100][0110][0111][1000][1001][1011][1100][1101][1110]
.
  [00] [00] [01] [01] [10] [10] [11] [11]
  [01] [10] [00] [11] [00] [11] [01] [10]
.
  [0] [0] [0] [0] [0] [0] [1] [1] [1] [1] [1] [1]
  [0] [0] [0] [1] [1] [1] [0] [0] [0] [1] [1] [1]
  [0] [1] [1] [0] [1] [1] [0] [0] [1] [0] [0] [1]
  [1] [0] [1] [0] [0] [1] [0] [1] [1] [0] [1] [0]

Mathematica

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}];
zaz[n_]:=Join@@(Table[Partition[#, d], {d, Divisors[n]}]&/@Tuples[{0, 1}, n]);
Table[Length[Select[zaz[n], apermatQ]], {n, 10}]

Crossrefs

Cf. A000740, A027375, A265627, A323351.

Cf. A323860, A323862, A323863, A323865, A323867, A323869.

Sequence in context: A148192 A373648 A192531 * A242659 A109388 A302919

Adjacent sequences: A323861 A323862 A323863 * A323865 A323866 A323867

Keyword

nonn

Author

Gus Wiseman, Feb 04 2019

Extensions

Terms a(18) and beyond from Andrew Howroyd, Aug 21 2019

Status

approved