A243611 Irregular triangular array of denominators of all rational numbers ordered as in Comments.

1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 3, 4, 3, 2, 1, 1, 3, 5, 5, 5, 3, 4, 3, 2, 1, 2, 3, 4, 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, 4, 7, 8, 7, 6, 5, 5, 5, 3, 4, 3, 2, 1, 2, 3, 4, 4, 7, 8, 7, 6, 5, 10, 13, 12, 11, 12, 13, 14

Offset: 1

Author

Clark Kimberling, Jun 08 2014

Keywords

• nonn
• easy
• tabf
• frac

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

Links

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

Crossrefs

Cf. A243612, A243613, A243614, A226130, 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]]