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

1,5

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.

Links

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

Example

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.

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 *)

Crossrefs

Cf. A243849, A243850, A243571.

Sequence in context: A341472 A047787 A102668 * A271617 A057741 A133571

Adjacent sequences: A243845 A243846 A243847 * A243849 A243850 A243851

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 12 2014

Status

approved