OFFSET
1,2
COMMENTS
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.
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
EXAMPLE
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
MATHEMATICA
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
(* Peter J. C. Moses, Feb 15 2011 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 10 2011
STATUS
approved