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

Author

Clark Kimberling, Feb 10 2011

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A057557, A186003, A186004.

Programs

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 *)