A186005 Distance array associated with ordering A057557 of N X N X N by antidiagonals (distances to xy plane).

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

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

Cf. A057557, A186003, A186004.

Sequence in context: A340579 A187566 A049831 * A080782 A116941 A166695

Adjacent sequences: A186002 A186003 A186004 * A186006 A186007 A186008

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 10 2011

Status

approved