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A243714 Irregular triangular array of denominators of all rational numbers ordered as in Comments.

%I A243714 #4 Jun 11 2014 21:19:55
%S A243714 1,1,1,2,1,1,3,2,1,1,4,3,2,1,2,3,5,4,3,2,1,3,5,5,6,3,5,4,3,2,1,1,4,7,
%T A243714 8,7,7,5,5,6,3,5,4,3,2,1,2,3,5,4,9,11,11,9,8,7,8,7,7,5,5,6,3,5,4,3,2,
%U A243714 1,3,5,5,6,7,8,11,7,14,15,14,11,9,4,9,11
%N A243714 Irregular triangular array of denominators of all rational numbers ordered as in Comments.
%C A243714 Let W denote the array of all positive rational numbers defined at A243712. For the present array, put (row 1) = (1), (row 2) = (-1, 3), (row 3) = (-1/2,0,3), and (row 4) = (-1/3,1/2,4).  Thereafter, (row n) consists of the following numbers in increasing order:  (row n) of W together -1/x for each x in (row n-1) of W.
%H A243714 Clark Kimberling, <a href="/A243714/b243714.txt">Table of n, a(n) for n = 1..2000</a>
%e A243714 First 6 rows of the array of all positive rationals:
%e A243714 1/1
%e A243714 -1/1 ... 2/1
%e A243714 -1/2 ... 0/1 ... 3/1
%e A243714 -1/3 ... 1/2 ... 4/1
%e A243714 -2/1 .... -1/4 ... 2/3 ... 3/2 ... 5/1
%e A243714 -3/2 ... -2/3 ... -1/5 ... 3/4 ... 5/3 ... 5/2 ... 6/1
%e A243714 The denominators, by rows:  1,1,1,2,1,1,3,2,1,1,4,3,2,1,2,3,5,4,3,2,1,...
%t A243714 z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -1/x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243714 h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, z}]; u1 = Delete[Flatten[u], 10]
%t A243714 w[1] = 0; w[2] = 1; w[3] = 1; w[n_] := w[n - 1] + w[n - 3];
%t A243714 u2 = Table[Drop[g[n], w[n]], {n, 1, z}];
%t A243714 u3 = Delete[Delete[Flatten[Map[Reverse, u2]], 4], 4]
%t A243714 Denominator[u3]  (* A243712 *)
%t A243714 Numerator[u3]    (* A243713 *)
%t A243714 Denominator[u1]  (* A243714 *)
%t A243714 Numerator[u1]    (* A243715 *)
%Y A243714 Cf. A243712, A243713, A243715, A000930, A226130, A243613.
%K A243714 nonn,easy,tabf,frac
%O A243714 1,4
%A A243714 _Clark Kimberling_, Jun 09 2014