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A184271
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Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal binary arrays (n >= 1, k >= 1).
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32
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2, 3, 3, 4, 7, 4, 6, 14, 14, 6, 8, 40, 64, 40, 8, 14, 108, 352, 352, 108, 14, 20, 362, 2192, 4156, 2192, 362, 20, 36, 1182, 14624, 52488, 52488, 14624, 1182, 36, 60, 4150, 99880, 699600, 1342208, 699600, 99880, 4150, 60, 108, 14602, 699252, 9587580, 35792568
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OFFSET
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1,1
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COMMENTS
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This is a 2-dimensional generalization of binary necklaces (A000031). A toroidal array or necklace can be defined either as an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns, or as a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns. - Gus Wiseman, Feb 04 2019
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LINKS
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FORMULA
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T(n,k) = (1/(nk))*Sum_{ c divides n } Sum_{ d divides k } phi(c)*phi(d)*2^(nk/lcm(c,d)), where phi is A000010 and lcm stands for least common multiple. - Stewart N. Ethier, Aug 24 2012
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EXAMPLE
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1 2 3 4 5 6 7
----------------------------------------------------------------------------
1: 2 3 4 6 8 14 20
2: 3 7 14 40 108 362 1182
3: 4 14 64 352 2192 14624 99880
4: 6 40 352 4156 52488 699600 9587580
5: 8 108 2192 52488 1342208 35792568 981706832
6: 14 362 14624 699600 35792568 1908897152 104715443852
7: 20 1182 99880 9587580 981706832 104715443852 11488774559744
8: 36 4150 699252 134223976 27487816992 5864063066500
9: 60 14602 4971184 1908881900 781874936816
10: 108 52588 35792568 27487869472
Inequivalent representatives of the T(2,3) = 14 toroidal necklaces:
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1]
[0 0 0] [0 0 1] [0 1 1] [1 1 1] [0 0 1] [0 1 0] [0 1 1]
.
[0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
[1 0 1] [1 1 0] [1 1 1] [0 1 1] [1 0 1] [1 1 1] [1 1 1]
(End)
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MATHEMATICA
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a[n_, k_] := Sum[If[Mod[n, c] == 0, Sum[If[Mod[k, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n k/LCM[c, d]), 0], {d, 1, k}], 0], {c, 1, n}]/(n k)
(* second program *)
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}]];
Table[Length[Select[Partition[#, n-k]&/@Tuples[{0, 1}, (n-k)*k], neckmatQ]], {n, 8}, {k, n-1}] (* Gus Wiseman, Feb 04 2019 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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