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A243712
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Irregular triangular array of denominators of all positive rational numbers ordered as in Comments.
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12
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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
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OFFSET
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1,5
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COMMENTS
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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 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.
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LINKS
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EXAMPLE
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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,...
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn,easy,tabf,frac
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AUTHOR
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STATUS
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approved
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