

A243848


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


6



1, 1, 1, 1, 3, 1, 3, 2, 1, 3, 2, 5, 5, 1, 3, 2, 5, 5, 3, 4, 3, 1, 3, 2, 5, 5, 3, 4, 7, 11, 5, 11, 7, 1, 3, 2, 5, 5, 3, 4, 7, 11, 5, 11, 7, 7, 7, 6, 8, 7, 7, 4, 1, 3, 2, 5, 5, 3, 4, 7, 11, 5, 11, 7, 7, 7, 6, 8, 7, 9, 17, 4, 9, 21, 17, 11, 5, 17, 21, 9, 17, 9
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OFFSET

1,5


COMMENTS

Decree that (row 1) = (1), (row 2) = (2), and (row 3) = (3). For n >= 4, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n1 together with 2/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. Let c(n) be the number of numbers in (row n); it appears that (c(n)) is not linearly recurrent.


LINKS

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


EXAMPLE

First 6 rows of the array of rationals:
1/1
2/1
3/1
4/1 ... 2/3
5/1 ... 5/3 ... 1/2
6/1 ... 8/3 ... 3/2 ... 6/5 ... 2/5
The denominators, by rows: 1,1,1,1,3,1,3,2,1,3,3,2,5,5.


MATHEMATICA

z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 2/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] (* A243848 *)
Numerator[v] (* A243849 *)
Table[Length[g[n]], {n, 1, z}] (* A243850 *)


CROSSREFS

Cf. A243849, A243850, A243571.
Sequence in context: A270390 A047787 A102668 * A271617 A057741 A133571
Adjacent sequences: A243845 A243846 A243847 * A243849 A243850 A243851


KEYWORD

nonn,easy,tabf,frac


AUTHOR

Clark Kimberling, Jun 12 2014


STATUS

approved



