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A243925 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

%I A243925 #5 Jun 19 2014 11:18:17
%S A243925 1,1,1,1,1,3,1,1,2,3,1,1,5,2,3,1,1,1,2,3,5,2,3,1,3,1,3,7,7,5,3,5,2,3,
%T A243925 1,3,4,5,5,4,7,7,5,3,5,2,3,1,1,5,7,5,13,7,13,9,5,4,7,7,7,5,3,5,2,3,1,
%U A243925 1,1,3,5,4,6,6,4,9,9,7,8,5,13,7,13,9,5
%N A243925 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.
%C A243925 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 A243925 Clark Kimberling, <a href="/A243925/b243925.txt">Table of n, a(n) for n = 1..2500</a>
%e A243925 First 7 rows of the array of rationals:
%e A243925 1/1
%e A243925 -2/1 ... 2/1
%e A243925 -1/1 ... 3/1
%e A243925 -2/3 ... 0/1 ... 4/1
%e A243925 -1/2 ... 1/3 ... 5/1
%e A243925 -6/1 ... -2/5 .. 1/2 ... 4/3 ... 6/1
%e A243925 -5/1 ... -4/1 .. -3/2 .. -1/3 .. 3/5 .. 3/2 .. 7/3 .. 7/1
%e A243925 The denominators, by rows:  1,1,1,1,1,3,1,1,2,3,1,1,5,2,3,1,1,1,2,3,5,2,3,1.
%t A243925 z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -2/x; h[1] = g[1];
%t A243925 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243925 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243925 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243925 u = Table[g[n], {n, 1, z}]
%t A243925 v = Delete[Flatten[u], 12]
%t A243925 Denominator[v] (* A243925 *)
%t A243925 Numerator[v]   (* A243926 *)
%Y A243925 Cf. A243926, A243927, A242359, A243712, A243928.
%K A243925 nonn,easy,tabf,frac
%O A243925 1,6
%A A243925 _Clark Kimberling_, Jun 15 2014