A186004 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to xz plane).

1, 2, 3, 4, 6, 7, 5, 9, 13, 14, 8, 12, 17, 24, 25, 10, 16, 23, 29, 40, 41, 11, 19, 28, 39, 46, 62, 63, 15, 22, 32, 45, 61, 69, 91, 92, 18, 27, 38, 50, 68, 90, 99, 128, 129, 20, 31, 44, 60, 74, 98, 127, 137, 174, 175, 21, 34, 49, 67, 89, 105, 136, 173, 184, 230, 231, 26, 37, 53, 73, 97, 126, 144, 183, 229, 241, 297, 298

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 A186004 lists those n for which j=n, the distance from (i,j,k) to the xz-plane. Every positive integer occurs exactly once in the array, so that as a sequence, A186004 is a permutation of the positive integers.

Examples

			Northwest corner:
   1,  2,  4,  5,  8, 10
   3,  6,  9, 12, 16, 19
   7, 13, 17, 23, 28, 32
  14, 24, 29, 39, 45, 50
  25, 40, 46, 61, 68, 74
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) < ...
		

Links

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

Crossrefs

Cf. A057557, A186003, A186005.

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[{3,12,2}];
ordering=lexicographicLattice[{2,Length[llha]}];
llha[[#1,#2]]&@@#1&/@ordering
(* Peter J. C. Moses, Feb 15 2011 *)