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

1,4

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.

Links

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

Example

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,...

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]]

Crossrefs

Cf. A243612, A243613, A243614, A226130, A000045.

Sequence in context: A229874 A364673 A330439 * A273102 A273104 A367680

Adjacent sequences: A243608 A243609 A243610 * A243612 A243613 A243614

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 08 2014

Status

approved