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A243854 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

%I A243854 #4 Jun 14 2014 21:41:42
%S A243854 1,1,1,1,1,1,3,5,1,3,5,3,1,3,5,9,7,3,7,1,3,5,9,7,3,5,7,7,5,2,1,3,5,9,
%T A243854 7,3,5,7,11,7,5,2,19,29,19,13,9,1,3,5,9,7,3,5,7,11,7,5,2,19,29,19,13,
%U A243854 11,17,9,9,17,11,13,19,6,4,5,1,3,5,9,7,3,5
%N A243854 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.
%C A243854 Decree that (row 1) = (1).  For n >= 2, row n consists of numbers in decreasing order generated as follows:  x+1 for each x in row n-1 together with 4/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 A243854 Clark Kimberling, <a href="/A243854/b243854.txt">Table of n, a(n) for n = 1..3000</a>
%e A243854 First 6 rows of the array of rationals:
%e A243854 1/1
%e A243854 4/1 ... 2/1
%e A243854 5/1 ... 3/1
%e A243854 6/1 ... 4/3 ... 4/5
%e A243854 7/1 ... 7/3 ... 9/5 ... 2/3
%e A243854 8/1 ... 10/3 ... 14/5 .. 20/9 .. 12/7 .. 5/3 .. 4/7
%e A243854 The denominators, by rows:  1,1,1,1,1,1,3,5,1,3,5,3,1,3,5,9,7,3,7.
%t A243854 z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 4/x; h[1] = g[1];
%t A243854 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243854 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243854 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243854 u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
%t A243854 Denominator[v] (* A243854 *)
%t A243854 Numerator[v]   (* A243855 *)
%t A243854 Table[Length[g[n]], {n, 1, z}] (* A243856 *)
%Y A243854 Cf. A243855, A243856, A242488, A243848.
%K A243854 nonn,easy,tabf,frac
%O A243854 1,7
%A A243854 _Clark Kimberling_, Jun 12 2014