

A243713


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


6



1, 2, 3, 4, 1, 5, 3, 2, 6, 5, 5, 3, 7, 7, 8, 7, 4, 1, 8, 9, 11, 11, 9, 4, 5, 3, 2, 9, 11, 14, 15, 14, 7, 11, 8, 7, 6, 5, 5, 3, 10, 13, 17, 19, 19, 10, 17, 13, 12, 13, 12, 13, 10, 7, 7, 8, 7, 4, 1, 11, 15, 20, 23, 24, 13, 23, 18, 17, 20, 19, 21, 17, 15, 16
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OFFSET

1,2


COMMENTS

Decree that (row 1) = (1), (row 2) = (3), 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 numerators, by rows: 1,2,3,4,1,5,3,2,6,5,5,3,7,7,8,7,4,1,8,9,11,11,9,4,5,3,2,...


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



