A323865 Number of aperiodic binary toroidal necklaces of size n.

1, 2, 2, 4, 8, 12, 36, 36, 114, 166, 396, 372, 1992, 1260, 4644, 8728, 20310, 15420, 87174, 55188, 314064, 399432, 762228, 729444, 5589620, 4026522, 10323180, 19883920, 57516048, 37025580, 286322136, 138547332, 805277760, 1041203944, 2021145660, 3926827224

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

Cf. A000031, A001037, A027375, A179043, A184271, A323351.

Cf. A323858, A323859, A323860, A323861, A323864, A323871.

Sequence in context: A153996 A123024 A079092 * A104700 A039941 A036761

Adjacent sequences: A323862 A323863 A323864 * A323866 A323867 A323868

Keyword

nonn

Author

Gus Wiseman, Feb 04 2019

Extensions

Terms a(19) and beyond from Andrew Howroyd, Aug 21 2019

Status

approved