A243713 Irregular triangular array of numerators of all positive rational numbers ordered as in Comments.

1, 2, 3, 4, 1, 5, 3, 2, 6, 5, 5, 3, 7, 7, 8, 7, 4, 1, 8, 9, 11, 11, 9, 4, 5, 3, 2, 9, 11, 14, 15, 14, 7, 11, 8, 7, 6, 5, 5, 3, 10, 13, 17, 19, 19, 10, 17, 13, 12, 13, 12, 13, 10, 7, 7, 8, 7, 4, 1, 11, 15, 20, 23, 24, 13, 23, 18, 17, 20, 19, 21, 17, 15, 16

Offset

1,2

Comments

Decree that (row 1) = (1), (row 2) = (3), and (row 3) = (3). Thereafter, row n consists of the following numbers arranged in decreasing order: 1+x for each x in (row n-1), together with x/(x + 1) for each x in row (n-3). Every positive rational number occurs exactly once in the array. The number of numbers in (row n) is A000930(n-1), for n >= 1.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

First 8 rows of the array of all positive rationals:
1/1
2/1
3/1
4/1 ... 1/2
5/1 ... 3/2 ... 2/3
6/1 ... 5/2 ... 5/3 ... 3/4
7/1 ... 7/2 ... 8/3 ... 7/4 ... 4/5 ... 1/3
8/1 ... 9/2 ... 11/3 .. 11/4 .. 9/5 ... 4/3 ... 5/6 ... 3/5 ... 2/5
The numerators, by rows:  1,2,3,4,1,5,3,2,6,5,5,3,7,7,8,7,4,1,8,9,11,11,9,4,5,3,2,...

Mathematica

z = 13; g[1] = {1}; f1[x_] := x + 1; 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}]; u1 = Delete[Flatten[u], 10]
w[1] = 0; w[2] = 1; w[3] = 1; w[n_] := w[n - 1] + w[n - 3];
u2 = Table[Drop[g[n], w[n]], {n, 1, z}];
u3 = Delete[Delete[Flatten[Map[Reverse, u2]], 4], 4]
Denominator[u3]  (* A243712 *)
Numerator[u3]    (* A243713 *)
Denominator[u1]  (* A243714 *)
Numerator[u1]    (* A243715 *)

Crossrefs

Cf. A243712, A243714, A243715, A000930, A243613.

Sequence in context: A115994 A276951 A071437 * A332266 A129709 A253146

Adjacent sequences: A243710 A243711 A243712 * A243714 A243715 A243716

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 09 2014

Status

approved