A243925 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

1, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 1, 5, 2, 3, 1, 1, 1, 2, 3, 5, 2, 3, 1, 3, 1, 3, 7, 7, 5, 3, 5, 2, 3, 1, 3, 4, 5, 5, 4, 7, 7, 5, 3, 5, 2, 3, 1, 1, 5, 7, 5, 13, 7, 13, 9, 5, 4, 7, 7, 7, 5, 3, 5, 2, 3, 1, 1, 1, 3, 5, 4, 6, 6, 4, 9, 9, 7, 8, 5, 13, 7, 13, 9, 5

Offset

1,6

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 -2/x for each nonzero x in row n-1, where duplicates are deleted as they occur. The number of numbers in row n is A243927(n). Conjecture: every rational number occurs exactly once in the array.

Links

Clark Kimberling, Table of n, a(n) for n = 1..2500

Example

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

Mathematica

z = 13; 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[g[n], {n, 1, z}]
v = Delete[Flatten[u], 12]
Denominator[v] (* A243925 *)
Numerator[v]   (* A243926 *)

Crossrefs

Cf. A243926, A243927, A242359, A243712, A243928.

Sequence in context: A336218 A016464 A245570 * A348282 A030727 A278564

Adjacent sequences: A243922 A243923 A243924 * A243926 A243927 A243928

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 15 2014

Status

approved