A243848 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

1, 1, 1, 1, 3, 1, 3, 2, 1, 3, 2, 5, 5, 1, 3, 2, 5, 5, 3, 4, 3, 1, 3, 2, 5, 5, 3, 4, 7, 11, 5, 11, 7, 1, 3, 2, 5, 5, 3, 4, 7, 11, 5, 11, 7, 7, 7, 6, 8, 7, 7, 4, 1, 3, 2, 5, 5, 3, 4, 7, 11, 5, 11, 7, 7, 7, 6, 8, 7, 9, 17, 4, 9, 21, 17, 11, 5, 17, 21, 9, 17, 9

Offset: 1

Author

Clark Kimberling, Jun 12 2014

Keywords

• nonn
• easy
• tabf
• frac

Comments

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.

Examples

			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 denominators, by rows:  1,1,1,1,3,1,3,2,1,3,3,2,5,5.
		

Links

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

Crossrefs

Cf. A243849, A243850, A243571.

Programs

Mathematica

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 *)