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Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.
7

%I #5 Jun 08 2014 09:17:31

%S 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 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 7,7,9,8,10,9,7,7,9,10,8,9,11,12,11,13

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

%C 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 Clark Kimberling, <a href="/A242359/b242359.txt">Table of n, a(n) for n = 1..2500</a>

%e First 6 rows of the array of rationals:

%e 1/1

%e 2/1

%e 3/1 ... 1/2

%e 4/1 ... 3/2 ... 1/3

%e 5/1 ... 5/2 ... 4/3 ... 2/3 ... 1/4

%e 6/1 ... 7/2 ... 7/3 ... 5/3 ... 5/4 ... 3/4 ... 2/5 ... 1/5

%e 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 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 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 Denominator[Flatten[t]] (* A242359 *)

%t Numerator[Flatten[t]] (* A242360 *)

%Y Cf. A226080, A242360, A000045.

%K nonn,easy,tabf,frac

%O 1,4

%A _Clark Kimberling_, Jun 07 2014