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Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
4

%I #4 Jun 14 2014 21:41:07

%S 1,2,3,4,2,5,5,1,6,8,3,6,2,7,11,5,11,7,4,3,1,8,14,7,16,12,7,7,10,10,4,

%T 6,2,9,17,9,21,17,10,11,17,21,9,17,9,8,6,5,5,4,3,1,10,20,11,26,22,13,

%U 15,24,32,14,28,16,15,13,11,13,11,14,22,5,10,22

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

%C Decree that (row 1) = (1), (row 2) = (2), and (row 3) = (3). For n >= 4, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n-1 together with 2/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. Let c(n) be the number of numbers in (row n); it appears that (c(n)) is not linearly recurrent.

%H Clark Kimberling, <a href="/A243849/b243849.txt">Table of n, a(n) for n = 1..3000</a>

%e First 6 rows of the array of rationals:

%e 1/1

%e 2/1

%e 3/1

%e 4/1 ... 2/3

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

%e 6/1 ... 8/3 ... 3/2 ... 6/5 ... 2/5

%e The numerators, by rows: 1,2,3,4,2,5,5,1,6,8,7,3,6,2.

%t z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 2/x; h[1] = g[1];

%t 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]];

%t g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]

%t u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];

%t Denominator[v] (* A243848 *)

%t Numerator[v] (* A243849 *)

%t Table[Length[g[n]], {n, 1, z}] (* A243850 *)

%Y Cf. A243848, A243850, A243571.

%K nonn,easy,tabf,frac

%O 1,2

%A _Clark Kimberling_, Jun 12 2014