A243733 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

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

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 = 5 and (row n) = h(n); the number of numbers in h(n) is A003520(n-1). (For m = 3, see A243712.)

Links

Clark Kimberling, Table of n, a(n) for n = 1..999

Example

First 11 rows of the array:
  1/1
  2/1
  3/1
  4/1
  5/1
  6/1 ... 1/2
  7/1 ... 3/2 ... 2/3
  8/1 ... 5/2 ... 5/3 ... 3/4
  9/1 ... 7/2 ... 8/3 ... 7/4 ... 4/5
  10/1 .. 9/2 ... 11/3 .. 11/4 .. 9/5 ... 5/6
  11/1 .. 11/2 .. 14/3 .. 15/4 .. 14/5 .. 11/6 .. 6/7 .. 1/3
The numerators, by rows:  1,2,3,4,5,6,1,7,3,2,8,5,5,3,9,7,8,7,4,10,9,...

Mathematica

z = 23; g[1] = {1}; g[2] = {2}; g[3] = {3}; g[4] = {4}; g[5] = {5};
g[n_] := Reverse[Union[1 + g[n - 1], g[n - 5]/(1 + g[n - 5])]]
Table[g[n], {n, 1, 9}]
v = Flatten[Table[g[n], {n, 1, z}]];
v1 = Denominator[v]; (* A243732 *)
v2 = Numerator[v];   (* A243733 *)

Crossrefs

Cf. A243732, A243713, A243731, A003520.

Sequence in context: A327241 A316913 A319092 * A334618 A138776 A379005

Adjacent sequences: A243730 A243731 A243732 * A243734 A243735 A243736

Keyword

nonn,easy,tabf,frac

Author

Clark Kimberling, Jun 09 2014

Status

approved