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

1,3

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.

Links

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

Example

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,...

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 *)

Crossrefs

Cf. A242363, A242359, A243574, A000045.

Sequence in context: A239957 A374684 A230040 * A116464 A346136 A284532

Adjacent sequences: A242358 A242359 A242360 * A242362 A242363 A242364

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 08 2014

Status

approved