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A243854
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Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.
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4
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1, 1, 1, 1, 1, 1, 3, 5, 1, 3, 5, 3, 1, 3, 5, 9, 7, 3, 7, 1, 3, 5, 9, 7, 3, 5, 7, 7, 5, 2, 1, 3, 5, 9, 7, 3, 5, 7, 11, 7, 5, 2, 19, 29, 19, 13, 9, 1, 3, 5, 9, 7, 3, 5, 7, 11, 7, 5, 2, 19, 29, 19, 13, 11, 17, 9, 9, 17, 11, 13, 19, 6, 4, 5, 1, 3, 5, 9, 7, 3, 5
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OFFSET
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1,7
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COMMENTS
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Decree that (row 1) = (1). For n >= 2, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n-1 together with 4/x for each x in row n-1, and duplicates are rejected as they occur. Every positive rational number occurs exactly once in the resulting array.
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LINKS
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EXAMPLE
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First 6 rows of the array of rationals:
1/1
4/1 ... 2/1
5/1 ... 3/1
6/1 ... 4/3 ... 4/5
7/1 ... 7/3 ... 9/5 ... 2/3
8/1 ... 10/3 ... 14/5 .. 20/9 .. 12/7 .. 5/3 .. 4/7
The denominators, by rows: 1,1,1,1,1,1,3,5,1,3,5,3,1,3,5,9,7,3,7.
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MATHEMATICA
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z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 4/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[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
Table[Length[g[n]], {n, 1, z}] (* A243856 *)
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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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