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%I #8 Aug 22 2019 22:19:14
%S 1,2,2,4,8,12,36,36,114,166,396,372,1992,1260,4644,8728,20310,15420,
%T 87174,55188,314064,399432,762228,729444,5589620,4026522,10323180,
%U 19883920,57516048,37025580,286322136,138547332,805277760,1041203944,2021145660,3926827224
%N Number of aperiodic binary toroidal necklaces of size n.
%C 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.
%H Andrew Howroyd, <a href="/A323865/b323865.txt">Table of n, a(n) for n = 0..200</a>
%H S. N. Ethier, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ethier/ethier2.html">Counting toroidal binary arrays</a>, J. Int. Seq. 16 (2013) #13.4.7.
%F a(n) = Sum_{d|n} A323861(d, n/d) for n > 0. - _Andrew Howroyd_, Aug 21 2019
%e Inequivalent representatives of the a(6) = 36 aperiodic necklaces:
%e 000001 000011 000101 000111 001011 001101 001111 010111 011111
%e .
%e 000 000 001 001 001 001 001 011 011
%e 001 011 010 011 101 110 111 101 111
%e .
%e 00 00 00 00 00 01 01 01 01
%e 00 01 01 01 11 01 01 10 11
%e 01 01 10 11 01 10 11 11 11
%e .
%e 0 0 0 0 0 0 0 0 0
%e 0 0 0 0 0 0 0 1 1
%e 0 0 0 0 1 1 1 0 1
%e 0 0 1 1 0 1 1 1 1
%e 0 1 0 1 1 0 1 1 1
%e 1 1 1 1 1 1 1 1 1
%t apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
%t neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
%t zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);
%t Table[If[n==0,1,Length[Union[First/@matcyc/@Select[zaz[n],And[apermatQ[#],neckmatQ[#]]&]]]],{n,0,10}]
%Y Cf. A000031, A001037, A027375, A179043, A184271, A323351.
%Y Cf. A323858, A323859, A323860, A323861, A323864, A323871.
%K nonn
%O 0,2
%A _Gus Wiseman_, Feb 04 2019
%E Terms a(19) and beyond from _Andrew Howroyd_, Aug 21 2019