A243929 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

1, -3, 2, -2, -3, 3, -1, -1, 3, 4, -3, 0, 1, 5, 5, 6, -6, -6, -3, 1, 7, 7, -12, -5, -6, -3, -1, 2, 5, 9, 8, -11, -15, -4, -12, -2, -3, 1, 4, 3, 4, 7, 9, 11, 9, 15, -21, -10, -13, -21, -15, -15, -7, -4, -6, -1, 3, 1, 5, 3, 8, 11, 8, 9, 12, 13, 13, 10, 16, -20

Offset: 1

Author

Clark Kimberling, Jun 15 2014

Keywords

• easy
• tabf
• frac
• sign

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 numerators, by rows:
  1,-3,2,-2,-3,3,-1,-1,3,4,-3,0,1,5,5,6,-6,6,-3,1,7,7,-12,-5,-6,-3,-1,2,5,9,8.
		

Links

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

Crossrefs

Cf. A243928, 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 *)