A243612 Irregular triangular array of numerators of all rational numbers ordered as in Comments.

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

Offset: 1

Author

Clark Kimberling, Jun 08 2014

Keywords

• easy
• tabf
• frac
• sign

Comments

Let F = A000045 (the Fibonacci numbers). Row n of the array to be generated consists of F(n-1) nonnegative rationals together with F(n-1) negative rationals. The nonnegatives, for n >=3, are x + 1 from the F(n-2) nonnegative numbers x in row n-1, together with x/(x + 1) from the F(n-3) nonnegative numbers x in row n-2. The negatives in row n are the negative reciprocals of the positives in row n.

Examples

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

Links

• Clark Kimberling, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A243611, A243613, A243614, A226131, A000045.

Programs

Mathematica

z = 12; g[1] = {0}; f1[x_] := x + 1; f2[x_] := -1/(x + 1); 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 = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 2, z}]
Delete[Flatten[Denominator[u]], 6]  (* A243611 *)
Delete[Flatten[Numerator[u]], 6]    (* A243612 *)
Delete[Flatten[Denominator[v]], 2]  (* A243613 *)
Delete[Flatten[Numerator[v]], 2]    (* A243614 *)
ListPlot[g[20]]