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A243926 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

%I A243926 #4 Jun 19 2014 11:18:28
%S A243926 1,-2,2,-1,3,-2,0,4,-1,1,5,-6,-2,1,4,6,-5,-4,-3,-1,3,3,7,7,-10,-3,-4,
%T A243926 -6,-2,2,2,8,5,10,8,-7,-5,-4,-3,-1,1,5,7,5,13,7,13,9,-14,-14,-10,-6,
%U A243926 -10,-4,-6,-2,1,3,6,8,12,12,8,18,9,16,10,-13,-10,-8,-9
%N A243926 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
%C A243926 Decree that (row 1) = (1).  For n >=2, row n consists of numbers in increasing order generated as follows:  x+1 for each x in row n-1 together with -2/x for each nonzero x in row n-1, where duplicates are deleted as they occur.  The number of numbers in row n is A243927(n).   Conjecture:  every rational number occurs exactly once in the array.
%H A243926 Clark Kimberling, <a href="/A243926/b243926.txt">Table of n, a(n) for n = 1..2500</a>
%e A243926 First 7 rows of the array of rationals:
%e A243926 1/1
%e A243926 -2/1 ... 2/1
%e A243926 -1/1 ... 3/1
%e A243926 -2/3 ... 0/1 ... 4/1
%e A243926 -1/2 ... 1/3 ... 5/1
%e A243926 -6/1 ... -2/5 .. 1/2 ... 4/3 ... 6/1
%e A243926 -5/1 ... -4/1 .. -3/2 .. -1/3 .. 3/5 .. 3/2 .. 7/3 .. 7/1
%e A243926 The numerators, by rows:  1,-2,2,-1,3,-2,0,4,-1,1,5,-6,-2,1,4,6,-5,-4,-3,-1,3,3,7,7.
%t A243926 z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -2/x; h[1] = g[1];
%t A243926 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243926 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243926 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243926 u = Table[g[n], {n, 1, z}]
%t A243926 v = Delete[Flatten[u], 12]
%t A243926 Denominator[v] (* A243925 *)
%t A243926 Numerator[v]   (* A243926 *)
%Y A243926 Cf. A243925, A243927.
%K A243926 easy,tabf,frac,sign
%O A243926 1,2
%A A243926 _Clark Kimberling_, Jun 15 2014