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

1,2

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 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.

Links

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

Example

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) < ...

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 *)

Crossrefs

Cf. A057557, A186004, A186005.

Sequence in context: A368258 A368264 A026166 * A368221 A077632 A347976

Adjacent sequences: A186000 A186001 A186002 * A186004 A186005 A186006

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 10 2011

Status

approved