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A186005
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Distance array associated with ordering A057557 of N X N X N by antidiagonals (distances to xy plane).
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4
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1, 3, 2, 4, 6, 5, 7, 8, 12, 11, 9, 13, 15, 22, 21, 10, 16, 23, 26, 37, 36, 14, 18, 27, 38, 42, 58, 57, 17, 24, 30, 43, 59, 64, 86, 85, 19, 28, 39, 47, 65, 87, 93, 122, 121, 20, 31, 44, 60, 70, 94, 123, 130, 167, 166, 25, 33, 48, 66, 88, 100, 131, 168, 176, 222, 221, 29, 40, 51, 71, 95, 124, 138, 177, 223, 232, 288, 287
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OFFSET
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1,2
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COMMENTS
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Let n=n(i,j,k) be the position of (i,j,k) in the lexicographic ordering A057557 of N X N X N, where N={1,2,3,...}. Row h of A186005 lists those n for which k=n, the distance from (i,j,k) to the xy-plane. Every positive integer occurs exactly once in the array, so that as a sequence, A186005 is a permutation of the positive integers.
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LINKS
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EXAMPLE
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T(2,2)=6, the position of (1,2,2) in the ordering
(1,1,1) < (1,1,2) < (1,2,1) < (2,1,1) < (1,1,3) < (1,2,2) < (1,3,1) < ...
Northwest corner:
1, 3, 4, 7, 9, 10
2, 6, 8, 13, 16, 18
5, 12, 15, 23, 27, 30
11, 22, 26, 38, 43, 47
21, 37, 42, 59, 65, 70
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MATHEMATICA
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lexicographicLattice[{dim_, maxHeight_}]:=Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1, {dim}], 1]&, maxHeight], 1];
lexicographicLatticeHeightArray[{dim_, maxHeight_, axis_}]:=Array[Flatten@Position[Map[#[[axis]]&, lexicographicLattice[{dim, maxHeight}]], #]&, maxHeight];
llha=lexicographicLatticeHeightArray[{3, 12, 3}];
ordering=lexicographicLattice[{2, Length[llha]}];
llha[[#1, #2]]&@@#1&/@ordering
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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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