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A186005 Distance array associated with ordering A057557 of N X N X N by antidiagonals (distances to xy plane). 4
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 (list; table; graph; refs; listen; history; text; internal format)
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: A129601 A187566 A049831 * A080782 A116941 A166695

Adjacent sequences:  A186002 A186003 A186004 * A186006 A186007 A186008

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 10 2011

STATUS

approved

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Last modified April 25 18:12 EDT 2019. Contains 322461 sequences. (Running on oeis4.)