A242363 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

1, 2, 1, 3, 2, 5, 3, 1, 3, 4, 8, 5, 2, 5, 3, 7, 13, 7, 8, 4, 1, 3, 4, 8, 5, 5, 11, 11, 21, 12, 7, 13, 7, 2, 5, 3, 7, 13, 7, 8, 4, 9, 18, 10, 19, 34, 18, 19, 9, 12, 21, 11, 11, 5, 1, 3, 4, 8, 5, 5, 11, 11, 21, 12, 7, 13, 7, 6, 14, 15, 29, 17, 14, 30, 29, 55

Offset

1,2

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 numerators, by rows:  1,2,1,3,2,5,3,1,3,4,8,5,2,5,3,7,13,7,8,4,...

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. A242361, A242359, A243574, A000045.

Sequence in context: A165006 A134735 A240867 * A050360 A175003 A176431

Adjacent sequences: A242360 A242361 A242362 * A242364 A242365 A242366

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 08 2014

Status

approved