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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A243614 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

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%I A243614 #5 Jun 11 2014 21:20:01
%S A243614 1,2,3,1,4,3,2,5,5,5,3,1,6,7,8,7,4,4,3,2,7,9,11,11,7,9,8,7,5,5,5,3,1,
%T A243614 8,11,14,15,10,14,13,12,11,12,13,10,5,6,7,8,7,4,4,3,2,9,13,17,19,13,
%U A243614 19,18,17,17,19,21,17,9,13,16,19,18,11,13,11,9,7
%N A243614 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
%C A243614 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 A243614 Clark Kimberling, <a href="/A243614/b243614.txt">Table of n, a(n) for n = 1..1500</a>
%e A243614 First 6 rows of the array of all positive rationals:
%e A243614 1/1
%e A243614 2/1
%e A243614 3/1 .. 1/2
%e A243614 4/1 .. 3/2 .. 2/3
%e A243614 5/1 .. 5/2 .. 5/3 .. 3/4 .. 1/3
%e A243614 6/1 .. 7/2 .. 8/3 .. 7/4 .. 4/3 .. 4/5 .. 3/5 .. 2/5
%e A243614 The numerators, by rows:  1,2,3,1,4,3,2,5,5,5,3,1,6,7,8,7,4,4,3,2...
%t A243614 z = 12; g[1] = {0}; f1[x_] := x + 1; f2[x_] := -1/(x + 1); h[1] = g[1];
%t A243614 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243614 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243614 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243614 u = Table[g[n], {n, 1, z}]
%t A243614 v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 2, z}]
%t A243614 Delete[Flatten[Denominator[u]], 6]  (* A243611 *)
%t A243614 Delete[Flatten[Numerator[u]], 6]    (* A243612 *)
%t A243614 Delete[Flatten[Denominator[v]], 2]  (* A243613 *)
%t A243614 Delete[Flatten[Numerator[v]], 2]    (* A243614 *)
%Y A243614 Cf. A243611, A243612, A243613, A000045.
%K A243614 nonn,easy,tabf,frac
%O A243614 1,2
%A A243614 _Clark Kimberling_, Jun 08 2014