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

1,5

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.

Links

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

Example

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.

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 *)

Crossrefs

Cf. A243929, A243930, A243925, A243712.

Sequence in context: A187450 A187449 A102541 * A286363 A233573 A025828

Adjacent sequences: A243925 A243926 A243927 * A243929 A243930 A243931

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 15 2014

Status

approved