%I #4 Jun 14 2014 21:41:20

%S 1,1,1,1,2,1,2,4,1,2,4,5,5,1,2,4,5,7,5,7,2,1,2,4,5,7,5,8,7,11,11,3,7,

%T 1,2,4,5,7,5,8,7,11,11,3,13,7,13,19,16,5,11,8,1,2,4,5,7,5,8,7,11,11,3,

%U 13,7,13,19,10,16,5,11,23,8,26,20,23,6,26,7

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

%C Decree that (row 1) = (1) and (row 2) = (3,2). For n >= 4, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n-1 together with 3/x for each x in row n-1, and duplicates are rejected as they occur. Every positive rational number occurs exactly once in the resulting array.

%H Clark Kimberling, <a href="/A243851/b243851.txt">Table of n, a(n) for n = 1..3000</a>

%e First 6 rows of the array of rationals:

%e 1/1

%e 3/1 ... 2/1

%e 4/1 ... 3/2

%e 5/1 ... 5/2 ... 3/4

%e 6/1 ... 7/2 ... 7/4 ... 6/5 ... 3/5

%e 7/1 ... 9/2 ... 11/4 .. 11/5 .. 12/7 .. 8/5 .. 6/7 .. 1/2

%e The denominators, by rows: 1,1,1,1,2,1,2,4,1,2,4,5,5,1,2,4,5,7,5,7,2.

%t z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 3/x; h[1] = g[1];

%t b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];

%t h[n_] := h[n] = Union[h[n - 1], g[n - 1]];

%t g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]

%t u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];

%t Denominator[v] (* A243851 *)

%t Numerator[v] (* A243852 *)

%t Table[Length[g[n]], {n, 1, z}] (* A243853 *)

%Y Cf. A243852, A243853, A242488.

%K nonn,easy,tabf,frac

%O 1,5

%A _Clark Kimberling_, Jun 12 2014