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A243851 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

%I A243851 #4 Jun 14 2014 21:41:20
%S A243851 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 A243851 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 A243851 13,7,13,19,10,16,5,11,23,8,26,20,23,6,26,7
%N A243851 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.
%C A243851 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 A243851 Clark Kimberling, <a href="/A243851/b243851.txt">Table of n, a(n) for n = 1..3000</a>
%e A243851 First 6 rows of the array of rationals:
%e A243851 1/1
%e A243851 3/1 ... 2/1
%e A243851 4/1 ... 3/2
%e A243851 5/1 ... 5/2 ... 3/4
%e A243851 6/1 ... 7/2 ... 7/4 ... 6/5 ... 3/5
%e A243851 7/1 ... 9/2 ... 11/4 .. 11/5 .. 12/7 .. 8/5 .. 6/7 .. 1/2
%e A243851 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 A243851 z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 3/x; h[1] = g[1];
%t A243851 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243851 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243851 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243851 u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
%t A243851 Denominator[v] (* A243851 *)
%t A243851 Numerator[v]   (* A243852 *)
%t A243851 Table[Length[g[n]], {n, 1, z}] (* A243853 *)
%Y A243851 Cf. A243852, A243853, A242488.
%K A243851 nonn,easy,tabf,frac
%O A243851 1,5
%A A243851 _Clark Kimberling_, Jun 12 2014