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

Author

Gus Wiseman, Feb 04 2019

Keywords

• nonn

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.

Examples

			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]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000740, A027375, A265627, A323351.

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

Programs

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}]

Formula

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

Extensions

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