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A243575 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

%I A243575 #4 Jun 19 2014 11:15:42
%S A243575 1,2,3,4,5,6,7,12,12,12,8,9,10,11,16,3,12,6,4,3,40,32,13,14,15,20,3,4,
%T A243575 6,12,15,21,19,56,26,16,11,68,52,17,18,19,24,1,12,2,12,15,21,24,9,30,
%U A243575 33,48,23,24,34,64,47,61,35,100,46,28,19,96,72,21,22
%N A243575 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
%C A243575 Decree that (row 1) = (1,2,3).  For n >=2, row n consists of numbers in increasing order generated as follows:  x+4 for each x in row n-1 together with 12/x for each nonzero x in row n-1, where duplicates are deleted as they occur.  Every rational number occurs exactly once in the array.  The number of numbers in row n is A022095(n-1) for n >= 4.
%H A243575 Clark Kimberling, <a href="/A243575/b243575.txt">Table of n, a(n) for n = 1..5000</a>
%e A243575 First 4 rows of the array of rationals:
%e A243575 1/1 .. 2/1 ... 3/1
%e A243575 4/1 .. 5/1 ... 6/2 . 7/1 . 12/1
%e A243575 12/7 . 12/5 .. 8/1 . 9/1 . 10/1 . 11/1 . 16/1
%e A243575 3/4 .. 12/11 . 6/5 . 4/3 . 3/2 .. 40/7 . 32/5 . 13/1 . 14/1 . 15/1 . 20/1
%e A243575 The numerators, by rows:  1,2,3,4,5,6,7,12,12,12,8,9,10,11,16,3,12,6,4,3,40,32,13,14,15,20.
%t A243575 z = 10; g[1] = {1, 2, 3}; f1[x_] := x + 4; f2[x_] := 12/x;  h[1] = g[1];
%t A243575 b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A243575 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A243575 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A243575 u = Table[g[n], {n, 1, z}]
%t A243575 v = Flatten[u]
%t A243575 Denominator[v]  (* A241837 *)
%t A243575 Numerator[v]    (* A243575 *)
%Y A243575 Cf. A241837, A243924, A022095.
%K A243575 nonn,easy,tabf,frac
%O A243575 1,2
%A A243575 _Clark Kimberling_, Jun 15 2014