A243928 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 4, 1, 2, 2, 1, 1, 1, 5, 5, 4, 2, 1, 1, 1, 7, 7, 5, 5, 4, 2, 1, 1, 2, 1, 5, 3, 8, 7, 7, 5, 5, 5, 4, 2, 1, 1, 1, 1, 2, 4, 4, 7, 5, 3, 11, 3, 11, 3, 8, 4, 7, 7, 5, 5, 5, 4, 2, 1, 1, 1, 1, 2, 5, 4, 4, 8, 11, 8, 3, 4, 7, 13, 13, 5

Offset: 1

Author

Clark Kimberling, Jun 15 2014

Keywords

• nonn
• easy
• tabf
• frac

Comments

Decree that (row 1) = (1). For n >=2, row n consists of numbers in increasing order generated as follows: x+1 for each x in row n-1 together with -3/x for each nonzero x in row n-1, where duplicates are deleted as they occur. The number of numbers in row n is A243930(n). Conjecture: every rational number occurs exactly once in the array.

Examples

			First 7 rows of the array of rationals:
1/1
-3/1 .. 2/1
-2/1 .. -3/2 .. 3/1
-1/1 .. -1/2 .. 3/2 ... 4/1
-3/4 .. 0/1 ... 1/2 ... 5/2 .. 5/1 .. 6/1
-6/1 .. -6/5 .. -3/5 .. 1/4 .. 7/2 .. 7/1
-12/1 . -5/1 .. -6/7 .. -3/7 . -1/5 . 2/5 . 5/4 . 9/2 . 8/1
The denominators, by rows:  1,1,1,1,2,1,1,2,2,1,4,1,2,2,1,1,1,5,5,4,2,1,1,1,7,7,5,5,4,2,1.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A243929, A243930, A243925, A243712.

Programs

Mathematica

z = 13; 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[g[n], {n, 1, z}]
v = Delete[Flatten[u], 23]
Denominator[v] (* A243928 *)
Numerator[v]   (* A243929 *)