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A243613
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Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.
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9
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1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 3, 1, 2, 3, 4, 3, 5, 5, 5, 1, 2, 3, 4, 3, 5, 5, 5, 6, 7, 8, 7, 4, 1, 2, 3, 4, 3, 5, 5, 5, 6, 7, 8, 7, 4, 7, 9, 11, 11, 7, 9, 8, 7, 1, 2, 3, 4, 3, 5, 5, 5, 6, 7, 8, 7, 4, 7, 9, 11, 11, 7, 9, 8, 7, 8, 11, 14, 15, 10, 14, 13, 12
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OFFSET
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1,4
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COMMENTS
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Let F = A000045 (the Fibonacci numbers). Decree that (row 1) = (1) and (row 2) = (2). Thereafter, row n consists of F(n) numbers in decreasing order, specifically, F(n-1) numbers x+1 from x in row n-1, together with F(n-2) numbers x/(x+1) from x in row n-2. The resulting array is also obtained by deleting from the array at A243611 all except the positive numbers and then reversing the rows.
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LINKS
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EXAMPLE
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First 6 rows of the array of all positive rationals:
1/1
2/1
3/1 .. 1/2
4/1 .. 3/2 .. 2/3
5/1 .. 5/2 .. 5/3 .. 3/4 .. 1/3
6/1 .. 7/2 .. 8/3 .. 7/4 .. 4/3 .. 4/5 .. 3/5 .. 2/5
The denominators, by rows: 1,1,1,2,1,2,3,1,2,3,4,3,1,2,3,4,3,5,5,5,...
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MATHEMATICA
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z = 12; g[1] = {0}; f1[x_] := x + 1; f2[x_] := -1/(x + 1); 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}]
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 2, z}]
Delete[Flatten[Denominator[u]], 6] (* A243611 *)
Delete[Flatten[Numerator[u]], 6] (* A243612 *)
Delete[Flatten[Denominator[v]], 2] (* A243613 *)
Delete[Flatten[Numerator[v]], 2] (* A243614 *)
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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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