A243715 Irregular triangular array of numerators of all rational numbers ordered as in Comments.

1, -1, 2, -1, 0, 3, -1, 1, 4, -2, -1, 2, 3, 5, -3, -2, -1, 3, 5, 5, 6, -4, -3, -2, -1, 1, 4, 7, 8, 7, 7, -3, -5, -4, -3, -2, -1, 2, 3, 5, 4, 9, 11, 11, 9, 8, -5, -5, -6, -3, -5, -4, -3, -2, -1, 3, 5, 5, 6, 7, 8, 11, 7, 14, 15, 14, 11, 9, -7, -8, -7, -7, -5

Offset: 1

Author

Clark Kimberling, Jun 09 2014

Keywords

• easy
• tabf
• frac
• sign

Comments

Let W denote the array of all positive rational numbers defined at A243712. For the present array, put (row 1) = (1), (row 2) = (-1, 3), (row 3) = (-1/2,0,3), and (row 4) = (-1/3,1/2,4). Thereafter, (row n) consists of the following numbers in increasing order: (row n) of W together -1/x for each x in (row n-1) of W.

Examples

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

Links

• Clark Kimberling, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A243712, A243713, A243714, A000930, A226131, A243613.

Programs

Mathematica

z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -1/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}]; u1 = Delete[Flatten[u], 10]
w[1] = 0; w[2] = 1; w[3] = 1; w[n_] := w[n - 1] + w[n - 3];
u2 = Table[Drop[g[n], w[n]], {n, 1, z}];
u3 = Delete[Delete[Flatten[Map[Reverse, u2]], 4], 4]
Denominator[u3]  (* A243712 *)
Numerator[u3]    (* A243713 *)
Denominator[u1]  (* A243714 *)
Numerator[u1]    (* A243715 *)