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A242359 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

%I A242359 #5 Jun 08 2014 09:17:31
%S A242359 1,1,1,2,1,2,3,1,2,3,3,4,1,2,3,3,4,4,5,5,1,2,3,3,4,4,5,5,5,5,7,7,6,1,
%T A242359 2,3,3,4,4,5,5,5,5,7,7,6,6,7,7,9,8,10,9,7,1,2,3,3,4,4,5,5,5,5,7,7,6,6,
%U A242359 7,7,9,8,10,9,7,7,9,10,8,9,11,12,11,13
%N A242359 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.
%C A242359 Decree that row 1 is (1) and row 2 is (2).  For n >=3, row n consists of numbers in decreasing order generated as follows:  x+1 for each x in row n-1 together with 1/(1+x) for each x in row n-2.  It is easy to prove that row n consists of F(n) numbers, where F = A000045 (the Fibonacci numbers), and that every positive rational number occurs exactly once.
%H A242359 Clark Kimberling, <a href="/A242359/b242359.txt">Table of n, a(n) for n = 1..2500</a>
%e A242359 First 6 rows of the array of rationals:
%e A242359 1/1
%e A242359 2/1
%e A242359 3/1 ... 1/2
%e A242359 4/1 ... 3/2 ... 1/3
%e A242359 5/1 ... 5/2 ... 4/3 ... 2/3 ... 1/4
%e A242359 6/1 ... 7/2 ... 7/3 ... 5/3 ... 5/4 ... 3/4 ... 2/5 ... 1/5
%e A242359 The denominators, by rows:  1,1,1,2,1,2,3,1,2,3,3,4,1,2,3,3,4,4,5,5.
%t A242359 z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 1/x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
%t A242359 h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; t = Table[Reverse[g[n]], {n, 1, z}]
%t A242359 Denominator[Flatten[t]] (* A242359 *)
%t A242359 Numerator[Flatten[t]]   (* A242360 *)
%Y A242359 Cf. A226080, A242360, A000045.
%K A242359 nonn,easy,tabf,frac
%O A242359 1,4
%A A242359 _Clark Kimberling_, Jun 07 2014