

A243852


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


4



1, 3, 2, 4, 3, 5, 5, 3, 6, 7, 7, 6, 3, 7, 9, 11, 11, 12, 8, 6, 1, 8, 11, 15, 16, 19, 13, 15, 13, 15, 12, 2, 3, 9, 13, 19, 21, 26, 18, 23, 20, 26, 23, 5, 21, 10, 15, 21, 15, 4, 6, 3, 10, 15, 23, 26, 33, 23, 31, 27, 37, 34, 8, 34, 17, 28, 40, 21, 31, 9, 17, 33
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OFFSET

1,2


COMMENTS

Decree that (row 1) = (1) and (row 2) = (3,2). For n >= 4, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n1 together with 3/x for each x in row n1, and duplicates are rejected as they occur. Every positive rational number occurs exactly once in the resulting array.


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..3000


EXAMPLE

First 6 rows of the array of rationals:
1/1
3/1 ... 2/1
4/1 ... 3/2
5/1 ... 5/2 ... 3/4
6/1 ... 7/2 ... 7/4 ... 6/5 ... 3/5
7/1 ... 9/2 ... 11/4 .. 11/5 .. 12/7 .. 8/5 .. 6/7 .. 1/2
The numerators, by rows: 1,3,2,4,3,5,5,3,6,7,7,6,3,7,9,11,11,12,8,6,1.


MATHEMATICA

z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 3/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[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
Denominator[v] (* A243851 *)
Numerator[v] (* A243852 *)
Table[Length[g[n]], {n, 1, z}] (* A243853 *)


CROSSREFS

Cf. A243851, A243853, A242488.
Sequence in context: A052938 A302391 A140114 * A025532 A195459 A133131
Adjacent sequences: A243849 A243850 A243851 * A243853 A243854 A243855


KEYWORD

nonn,easy,tabf,frac


AUTHOR

Clark Kimberling, Jun 12 2014


STATUS

approved



