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

%I A186003 #17 Jul 25 2017 02:28:44
%S A186003 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,
%T A186003 82,84,12,26,32,52,79,83,118,120,13,27,47,53,80,115,119,163,165,14,28,
%U A186003 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
%N A186003 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to yz plane).
%C A186003 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.
%H A186003 G. C. Greubel, <a href="/A186003/b186003.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e A186003 Northwest corner:
%e A186003    1,  2,  3,  5,  6,  7,  11
%e A186003    4,  8,  9, 15, 16, 17,  26
%e A186003   10, 18, 19, 30, 31, 32,  47
%e A186003   20, 33, 34, 51, 52, 53,  75
%e A186003   35, 54, 55, 79, 80, 81, 111
%e A186003 T(2,1)=4, the position of (2,1,1) in the ordering
%e A186003 (1,1,1) < (1,1,2) < (1,2,1) < (2,1,1) < (1,1,3) < (1,2,2) < (1,3,1) < ...
%t A186003 lexicographicLattice[{dim_,maxHeight_}]:=Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1];
%t A186003 lexicographicLatticeHeightArray[{dim_,maxHeight_,axis_}]:=Array[Flatten@Position[Map[#[[axis]]&,lexicographicLattice[{dim,maxHeight}]],#]&,maxHeight];
%t A186003 llha=lexicographicLatticeHeightArray[{3,12,1}];
%t A186003 ordering=lexicographicLattice[{2,Length[llha]}];
%t A186003 llha[[#1,#2]]&@@#1&/@ordering
%t A186003 (* _Peter J. C. Moses_, Feb 15 2011 *)
%Y A186003 Cf. A057557, A186004, A186005.
%K A186003 nonn,tabl
%O A186003 1,2
%A A186003 _Clark Kimberling_, Feb 10 2011