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A186005 Distance array associated with ordering A057557 of N X N X N by antidiagonals (distances to xy plane).

%I A186005 #16 Jul 25 2017 02:28:16
%S A186005 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,
%T A186005 58,57,17,24,30,43,59,64,86,85,19,28,39,47,65,87,93,122,121,20,31,44,
%U A186005 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
%N A186005 Distance array associated with ordering A057557 of N X N X N by antidiagonals (distances to xy plane).
%C A186005 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.
%H A186005 G. C. Greubel, <a href="/A186005/b186005.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e A186005 T(2,2)=6, the position of (1,2,2) in the ordering
%e A186005 (1,1,1) < (1,1,2) < (1,2,1) < (2,1,1) < (1,1,3) < (1,2,2) < (1,3,1) < ...
%e A186005 Northwest corner:
%e A186005    1,  3,  4,  7,  9, 10
%e A186005    2,  6,  8, 13, 16, 18
%e A186005    5, 12, 15, 23, 27, 30
%e A186005   11, 22, 26, 38, 43, 47
%e A186005   21, 37, 42, 59, 65, 70
%t A186005 lexicographicLattice[{dim_,maxHeight_}]:=Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1];
%t A186005 lexicographicLatticeHeightArray[{dim_,maxHeight_,axis_}]:=Array[Flatten@Position[Map[#[[axis]]&,lexicographicLattice[{dim,maxHeight}]],#]&,maxHeight];
%t A186005 llha=lexicographicLatticeHeightArray[{3,12,3}];
%t A186005 ordering=lexicographicLattice[{2,Length[llha]}];
%t A186005 llha[[#1,#2]]&@@#1&/@ordering
%t A186005 (* _Peter J. C. Moses_, Feb 15 2011 *)
%Y A186005 Cf. A057557, A186003, A186004.
%K A186005 nonn,tabl
%O A186005 1,2
%A A186005 _Clark Kimberling_, Feb 10 2011