A186003 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to yz plane).

1, 2, 4, 3, 8, 10, 5, 9, 18, 20, 6, 15, 19, 33, 35, 7, 16, 30, 34, 54, 56, 11, 17, 31, 51, 55, 82, 84, 12, 26, 32, 52, 79, 83, 118, 120, 13, 27, 47, 53, 80, 115, 119, 163, 165, 14, 28, 48, 75, 81, 116, 160, 164, 218, 220, 21, 29, 49, 76, 111, 117, 161, 215, 219, 284, 286, 22, 42, 50, 77, 112, 156, 162, 216, 281, 285, 362, 364

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 A186003 lists those n for which i=n, the distance from (i,j,k) to the yz-plane. Every positive integer occurs exactly once in the array, so that as a sequence, A186003 is a permutation of the positive integers.

Examples

			Northwest corner:
   1,  2,  3,  5,  6,  7,  11
   4,  8,  9, 15, 16, 17,  26
  10, 18, 19, 30, 31, 32,  47
  20, 33, 34, 51, 52, 53,  75
  35, 54, 55, 79, 80, 81, 111
T(2,1)=4, the position of (2,1,1) 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, A186004, 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[{3,12,1}];
ordering=lexicographicLattice[{2,Length[llha]}];
llha[[#1,#2]]&@@#1&/@ordering
(* Peter J. C. Moses, Feb 15 2011 *)