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A243852 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

%I A243852 #5 Jun 14 2014 21:41:29
%S A243852 1,3,2,4,3,5,5,3,6,7,7,6,3,7,9,11,11,12,8,6,1,8,11,15,16,19,13,15,13,
%T A243852 15,12,2,3,9,13,19,21,26,18,23,20,26,23,5,21,10,15,21,15,4,6,3,10,15,
%U A243852 23,26,33,23,31,27,37,34,8,34,17,28,40,21,31,9,17,33
%N A243852 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
%C A243852 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 A243852 Clark Kimberling, <a href="/A243852/b243852.txt">Table of n, a(n) for n = 1..3000</a>
%e A243852 First 6 rows of the array of rationals:
%e A243852 1/1
%e A243852 3/1 ... 2/1
%e A243852 4/1 ... 3/2
%e A243852 5/1 ... 5/2 ... 3/4
%e A243852 6/1 ... 7/2 ... 7/4 ... 6/5 ... 3/5
%e A243852 7/1 ... 9/2 ... 11/4 .. 11/5 .. 12/7 .. 8/5 .. 6/7 .. 1/2
%e A243852 The numerators, by rows:  1,3,2,4,3,5,5,3,6,7,7,6,3,7,9,11,11,12,8,6,1.
%t A243852 z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 3/x; h[1] = g[1];
%t A243852 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243852 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243852 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243852 u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
%t A243852 Denominator[v] (* A243851 *)
%t A243852 Numerator[v]   (* A243852 *)
%t A243852 Table[Length[g[n]], {n, 1, z}] (* A243853 *)
%Y A243852 Cf. A243851, A243853, A242488.
%K A243852 nonn,easy,tabf,frac
%O A243852 1,2
%A A243852 _Clark Kimberling_, Jun 12 2014