A243714 Irregular triangular array of denominators of all rational numbers ordered as in Comments.

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

Offset: 1

Author

Clark Kimberling, Jun 09 2014

Keywords

• nonn
• easy
• tabf
• frac

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 denominators, by rows:  1,1,1,2,1,1,3,2,1,1,4,3,2,1,2,3,5,4,3,2,1,...
		

Links

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

Crossrefs

Cf. A243712, A243713, A243715, A000930, A226130, 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 *)