cp's OEIS Frontend

A323865 Number of aperiodic binary toroidal necklaces of size n.

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