A243926 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

1, -2, 2, -1, 3, -2, 0, 4, -1, 1, 5, -6, -2, 1, 4, 6, -5, -4, -3, -1, 3, 3, 7, 7, -10, -3, -4, -6, -2, 2, 2, 8, 5, 10, 8, -7, -5, -4, -3, -1, 1, 5, 7, 5, 13, 7, 13, 9, -14, -14, -10, -6, -10, -4, -6, -2, 1, 3, 6, 8, 12, 12, 8, 18, 9, 16, 10, -13, -10, -8, -9

Offset

1,2

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 numerators, by rows:  1,-2,2,-1,3,-2,0,4,-1,1,5,-6,-2,1,4,6,-5,-4,-3,-1,3,3,7,7.

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. A243925, A243927.

Sequence in context: A069004 A182490 A053274 * A281013 A190683 A181810

Adjacent sequences: A243923 A243924 A243925 * A243927 A243928 A243929

Keyword

easy,tabf,frac,sign

Author

Clark Kimberling, Jun 15 2014

Status

approved