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.
%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