OFFSET
0,2
COMMENTS
We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. 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
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.
FORMULA
a(n) = Sum_{d|n} A323861(d, n/d) for n > 0. - Andrew Howroyd, Aug 21 2019
EXAMPLE
Inequivalent representatives of the a(6) = 36 aperiodic necklaces:
000001 000011 000101 000111 001011 001101 001111 010111 011111
.
000 000 001 001 001 001 001 011 011
001 011 010 011 101 110 111 101 111
.
00 00 00 00 00 01 01 01 01
00 01 01 01 11 01 01 10 11
01 01 10 11 01 10 11 11 11
.
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 1
0 0 0 0 1 1 1 0 1
0 0 1 1 0 1 1 1 1
0 1 0 1 1 0 1 1 1
1 1 1 1 1 1 1 1 1
MATHEMATICA
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@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[If[n==0, 1, Length[Union[First/@matcyc/@Select[zaz[n], And[apermatQ[#], neckmatQ[#]]&]]]], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2019
EXTENSIONS
Terms a(19) and beyond from Andrew Howroyd, Aug 21 2019
STATUS
approved