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
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