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A243849 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

%I A243849 #4 Jun 14 2014 21:41:07
%S A243849 1,2,3,4,2,5,5,1,6,8,3,6,2,7,11,5,11,7,4,3,1,8,14,7,16,12,7,7,10,10,4,
%T A243849 6,2,9,17,9,21,17,10,11,17,21,9,17,9,8,6,5,5,4,3,1,10,20,11,26,22,13,
%U A243849 15,24,32,14,28,16,15,13,11,13,11,14,22,5,10,22
%N A243849 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
%C A243849 Decree that (row 1) = (1), (row 2) = (2), and (row 3) = (3).  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 2/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.  Let c(n) be the number of numbers in (row n); it appears that (c(n)) is not linearly recurrent.
%H A243849 Clark Kimberling, <a href="/A243849/b243849.txt">Table of n, a(n) for n = 1..3000</a>
%e A243849 First 6 rows of the array of rationals:
%e A243849 1/1
%e A243849 2/1
%e A243849 3/1
%e A243849 4/1 ... 2/3
%e A243849 5/1 ... 5/3 ... 1/2
%e A243849 6/1 ... 8/3 ... 3/2 ... 6/5 ... 2/5
%e A243849 The numerators, by rows:  1,2,3,4,2,5,5,1,6,8,7,3,6,2.
%t A243849 z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 2/x; h[1] = g[1];
%t A243849 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243849 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243849 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243849 u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
%t A243849 Denominator[v] (* A243848 *)
%t A243849 Numerator[v]   (* A243849 *)
%t A243849 Table[Length[g[n]], {n, 1, z}] (* A243850 *)
%Y A243849 Cf. A243848, A243850, A243571.
%K A243849 nonn,easy,tabf,frac
%O A243849 1,2
%A A243849 _Clark Kimberling_, Jun 12 2014