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A243613 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

%I A243613 #7 Jun 11 2014 21:19:48
%S A243613 1,1,1,2,1,2,3,1,2,3,4,3,1,2,3,4,3,5,5,5,1,2,3,4,3,5,5,5,6,7,8,7,4,1,
%T A243613 2,3,4,3,5,5,5,6,7,8,7,4,7,9,11,11,7,9,8,7,1,2,3,4,3,5,5,5,6,7,8,7,4,
%U A243613 7,9,11,11,7,9,8,7,8,11,14,15,10,14,13,12
%N A243613 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.
%C A243613 Let F = A000045 (the Fibonacci numbers).  Decree that (row 1) = (1) and (row 2) = (2).  Thereafter, row n consists of F(n) numbers in decreasing order, specifically, F(n-1) numbers x+1 from x in row n-1, together with F(n-2) numbers x/(x+1) from x in row n-2.  The resulting array is also obtained by deleting from the array at A243611 all except the positive numbers and then reversing the rows.
%H A243613 Clark Kimberling, <a href="/A243613/b243613.txt">Table of n, a(n) for n = 1..1500</a>
%e A243613 First 6 rows of the array of all positive rationals:
%e A243613 1/1
%e A243613 2/1
%e A243613 3/1 .. 1/2
%e A243613 4/1 .. 3/2 .. 2/3
%e A243613 5/1 .. 5/2 .. 5/3 .. 3/4 .. 1/3
%e A243613 6/1 .. 7/2 .. 8/3 .. 7/4 .. 4/3 .. 4/5 .. 3/5 .. 2/5
%e A243613 The denominators, by rows:  1,1,1,2,1,2,3,1,2,3,4,3,1,2,3,4,3,5,5,5,...
%t A243613 z = 12; g[1] = {0}; f1[x_] := x + 1; f2[x_] := -1/(x + 1); h[1] = g[1];
%t A243613 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243613 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243613 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243613 u = Table[g[n], {n, 1, z}]
%t A243613 v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 2, z}]
%t A243613 Delete[Flatten[Denominator[u]], 6]  (* A243611 *)
%t A243613 Delete[Flatten[Numerator[u]], 6]    (* A243612 *)
%t A243613 Delete[Flatten[Denominator[v]], 2]  (* A243613 *)
%t A243613 Delete[Flatten[Numerator[v]], 2]    (* A243614 *)
%Y A243613 Cf. A243611, A243612, A243614, A000045.
%K A243613 nonn,easy,tabf,frac
%O A243613 1,4
%A A243613 _Clark Kimberling_, Jun 08 2014