A242361 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

1, 1, 2, 2, 3, 3, 1, 3, 5, 3, 5, 2, 5, 8, 4, 5, 8, 4, 3, 1, 4, 8, 7, 13, 7, 4, 8, 7, 13, 7, 3, 5, 2, 7, 13, 7, 12, 21, 11, 11, 5, 7, 13, 7, 12, 21, 11, 11, 5, 5, 8, 4, 3, 1, 5, 11, 11, 21, 12, 9, 19, 18, 34, 19, 10, 18, 9, 5, 11, 11, 21, 12, 9, 19, 18, 34

Offset: 1

Author

Clark Kimberling, Jun 08 2014

Keywords

• nonn
• easy
• tabf
• frac

Comments

Let F = A000045 (the Fibonacci numbers). To construct the array of positive rationals, decree that row 1 is (1) and row 2 is (2). Thereafter, row n consists of the following numbers in increasing order: the F(n-2) numbers 1/x from numbers x > 1 in row n-1, together with the F(n-3) numbers 1 + 1/x from numbers x < 1 in row n - 1, together with the F(n - 2) numbers (2*x + 1)/ (x + 1) from numbers x in row n-2. Row n consists of F(n) numbers ranging from 1/((n+1)/2) to n/2 if n is odd and from 2/(n-1) to (n+2)/2 if n is even.

Examples

			First 6 rows of the array of rationals:
1/1
2/1
1/2 ... 3/2
2/3 ... 5/3 ... 3/1
1/3 ... 3/5 ... 4/3 ... 8/5 ... 5/2
2/5 ... 5/8 ... 3/4 ... 7/5 ... 13/8 .. 7/4 ... 8/3 ... 4/1
The denominators, by rows:  1,1,2,2,3,3,1,3,5,3,5,2,5,8,4,5,8,4,3,1,...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A242363, A242359, A243574, A000045.

Programs

Mathematica

z = 18; g[1] = {1}; f1[x_] := 1 + 1/x; 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}]; v = Flatten[u]; Length[v]
Denominator[v];  (* A242361 *)
Numerator[v];    (* A242363 *)