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A323865
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Number of aperiodic binary toroidal necklaces of size n.
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14
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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Inequivalent representatives of the a(6) = 36 aperiodic necklaces:
000001 000011 000101 000111 001011 001101 001111 010111 011111
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000 000 001 001 001 001 001 011 011
001 011 010 011 101 110 111 101 111
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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
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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
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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