1,2

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.

Clark Kimberling, Table of n, a(n) for n = 1..3000

First 6 rows of the array of rationals:

1/1

2/1

3/1

4/1 ... 2/3

5/1 ... 5/3 ... 1/2

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

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

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

b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];

h[n_] := h[n] = Union[h[n - 1], g[n - 1]];

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

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

Denominator[v] (* A243848 *)

Numerator[v] (* A243849 *)

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

Cf. A243848, A243850, A243571.

Sequence in context: A260643 A366295 A300840 * A286547 A157000 A380300

Adjacent sequences: A243846 A243847 A243848 * A243850 A243851 A243852

nonn,easy,tabf,frac

Clark Kimberling, Jun 12 2014

approved