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A242363 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

%I A242363 #5 Jun 11 2014 21:19:05
%S A242363 1,2,1,3,2,5,3,1,3,4,8,5,2,5,3,7,13,7,8,4,1,3,4,8,5,5,11,11,21,12,7,
%T A242363 13,7,2,5,3,7,13,7,8,4,9,18,10,19,34,18,19,9,12,21,11,11,5,1,3,4,8,5,
%U A242363 5,11,11,21,12,7,13,7,6,14,15,29,17,14,30,29,55
%N A242363 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
%C A242363 Let F = A000045 (the Fibonacci numbers).  To construct the array of positive rationals, decree that row 1 is (1) and row 2 is (2).   Thereafter, row n consists of the following numbers in increasing order: the F(n-2) numbers 1/x from numbers x > 1 in row n-1, together with the F(n-3) numbers 1 + 1/x from numbers x < 1 in row n - 1, together with the F(n - 2) numbers (2*x + 1)/ (x + 1) from numbers x in row n-2.  Row n consists of F(n) numbers ranging from 1/((n+1)/2) to n/2 if n is odd and from 2/(n-1) to (n+2)/2 if n is even.
%H A242363 Clark Kimberling, <a href="/A242363/b242363.txt">Table of n, a(n) for n = 1..5000</a>
%e A242363 First 6 rows of the array of rationals:
%e A242363 1/1
%e A242363 2/1
%e A242363 1/2 ... 3/2
%e A242363 2/3 ... 5/3 ... 3/1
%e A242363 1/3 ... 3/5 ... 4/3 ... 8/5 ... 5/2
%e A242363 2/5 ... 5/8 ... 3/4 ... 7/5 ... 13/8 .. 7/4 ... 8/3 ... 4/1
%e A242363 The numerators, by rows:  1,2,1,3,2,5,3,1,3,4,8,5,2,5,3,7,13,7,8,4,...
%t A242363 z = 18; g[1] = {1}; f1[x_] := 1 + 1/x; f2[x_] := 1/x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A242363 h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
%t A242363 g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
%t A242363 u = Table[g[n], {n, 1, z}]; v = Flatten[u]; Length[v]
%t A242363 Denominator[v];  (* A242361 *)
%t A242363 Numerator[v];    (* A242363 *)
%Y A242363 Cf. A242361, A242359, A243574, A000045.
%K A242363 nonn,easy,tabf,frac
%O A242363 1,2
%A A242363 _Clark Kimberling_, Jun 08 2014