A243851 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

1, 1, 1, 1, 2, 1, 2, 4, 1, 2, 4, 5, 5, 1, 2, 4, 5, 7, 5, 7, 2, 1, 2, 4, 5, 7, 5, 8, 7, 11, 11, 3, 7, 1, 2, 4, 5, 7, 5, 8, 7, 11, 11, 3, 13, 7, 13, 19, 16, 5, 11, 8, 1, 2, 4, 5, 7, 5, 8, 7, 11, 11, 3, 13, 7, 13, 19, 10, 16, 5, 11, 23, 8, 26, 20, 23, 6, 26, 7

Offset

1,5

Comments

Decree that (row 1) = (1) and (row 2) = (3,2). 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 3/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.

Links

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

Example

First 6 rows of the array of rationals:
1/1
3/1 ... 2/1
4/1 ... 3/2
5/1 ... 5/2 ... 3/4
6/1 ... 7/2 ... 7/4 ... 6/5 ... 3/5
7/1 ... 9/2 ... 11/4 .. 11/5 .. 12/7 .. 8/5 .. 6/7 .. 1/2
The denominators, by rows:  1,1,1,1,2,1,2,4,1,2,4,5,5,1,2,4,5,7,5,7,2.

Mathematica

z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 3/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] (* A243851 *)
Numerator[v]   (* A243852 *)
Table[Length[g[n]], {n, 1, z}] (* A243853 *)

Crossrefs

Cf. A243852, A243853, A242488.

Sequence in context: A141539 A340547 A327844 * A168266 A059250 A303696

Adjacent sequences: A243848 A243849 A243850 * A243852 A243853 A243854

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 12 2014

Status

approved