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Decree that (row 1) = (1), (row 2) = (32), and (row 3) = (3). Thereafter, row n consists of the following numbers arranged in decreasing order: 1 + x for each x in (row n-1), together with x/(x + 1) for each x in row (n-3). Every positive rational number occurs exactly once in the array. The number of numbers in (row n) is A000930(n-1), for n >= 1.

allocated for Clark KimberlingIrregular triangular array of denominators of all positive rational numbers ordered as in Comments.

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

1,5

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 n-1), together with x/(x + 1) for each x in row (n-3). Every positive rational number occurs exactly once in the array. The number of numbers in (row n) is A000930(n-1), for n >= 1.

Clark Kimberling, <a href="/A243712/b243712.txt">Table of n, a(n) for n = 1..1000</a>

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,...

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]

Denominator[u3] (* A243712 *)

Numerator[u3] (* A243713 *)

Denominator[u1] (* A243714 *)

Numerator[u1] (* A243715 *)

Cf. A243713, A243714, A243715, A000930, A243613.

allocated

nonn,easy,tabf,frac

Clark Kimberling, Jun 09 2014

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