

A243712


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


12



1, 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 3, 1, 2, 3, 4, 5, 3, 6, 5, 5, 1, 2, 3, 4, 5, 3, 6, 5, 5, 7, 7, 8, 7, 1, 2, 3, 4, 5, 3, 6, 5, 5, 7, 7, 8, 7, 8, 9, 11, 11, 9, 4, 1, 2, 3, 4, 5, 3, 6, 5, 5, 7, 7, 8, 7, 8, 9, 11, 11, 9, 4, 9, 11, 14, 15, 14, 7
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,5


COMMENTS

Decree that (row 1) = (1), (row 2) = (2), and (row 3) = (3). Thereafter, row n consists of the following numbers arranged in decreasing order: 1 + x for each x in (row n1), together with x/(x + 1) for each x in row (n3). Every positive rational number occurs exactly once in the array. The number of numbers in (row n) is A000930(n1), for n >= 1.


LINKS



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 denominators, by rows: 1,1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,3,1,2,3,4,5,3,6,5,5,...


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]


CROSSREFS



KEYWORD

nonn,easy,tabf,frac


AUTHOR



STATUS

approved



