A243731 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

1, 2, 3, 4, 5, 1, 6, 3, 2, 7, 5, 5, 3, 8, 7, 8, 7, 4, 9, 9, 11, 11, 9, 5, 1, 10, 11, 14, 15, 14, 11, 4, 6, 3, 2, 11, 13, 17, 19, 19, 17, 7, 13, 8, 7, 7, 5, 5, 3, 12, 15, 20, 23, 24, 23, 10, 20, 13, 12, 15, 12, 13, 10, 8, 7, 8, 7, 4, 13, 17, 23, 27, 29, 29

Offset

1,2

Comments

Suppose that m >= 3, and define sets h(n) of positive rational numbers as follows: h(n) = {n} for n = 1..m, and thereafter, h(n) = Union({x+1: x in h(n-1)}, {x/(x+1) : x in h(n-m)}), with the numbers in h(n) arranged in decreasing order. Every positive rational lies in exactly one of the sets h(n). For the present array, put m = 4 and (row n) = h(n); the number of numbers in h(n) is A003269(n-1). (For m = 3, see A243712.)

Links

Clark Kimberling, Table of n, a(n) for n = 1..6000

Example

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

Mathematica

z = 27; g[1] = {1}; g[2] = {2}; g[3] = {3}; g[4] = {4};
g[n_] := Reverse[Union[1 + g[n - 1], g[n - 4]/(1 + g[n - 4])]]
Table[g[n], {n, 1, 12}]
v = Flatten[Table[g[n], {n, 1, z}]];
Denominator[v];  (* A243730 *)
Numerator[v];  (* A243731 *)

Crossrefs

Cf. A243730, A243712, A003269.

Sequence in context: A026284 A071489 A071514 * A334945 A273825 A082299

Adjacent sequences: A243728 A243729 A243730 * A243732 A243733 A243734

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 09 2014

Status

approved