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A243733
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Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.
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3
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1, 2, 3, 4, 5, 6, 1, 7, 3, 2, 8, 5, 5, 3, 9, 7, 8, 7, 4, 10, 9, 11, 11, 9, 5, 11, 11, 14, 15, 14, 11, 6, 1, 12, 13, 17, 19, 19, 17, 13, 4, 7, 3, 2, 13, 15, 20, 23, 24, 23, 20, 7, 15, 8, 7, 8, 5, 5, 3, 14, 17, 23, 27, 29, 29, 27, 10, 23, 13, 12, 17, 12, 13
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OFFSET
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1,2
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COMMENTS
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Suppose that m >= 3, and define sets h(n) of positive rational numbers as follows: h(n) = {n} for n = 1..m, and thereafter, h(n) = Union({x+1: x in h(n-1), {x/(x+1) : x in h(n-m)}), with the numbers in h(n) arranged in decreasing order. Every positive rational lies in exactly one of the sets h(n). For the present array, put m = 5 and (row n) = h(n); the number of numbers in h(n) is A003520(n-1). (For m = 3, see A243712.)
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LINKS
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EXAMPLE
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First 9 rows of the array:
1/1
2/1
3/1
4/1
5/1
6/1 ... 1/2
7/1 ... 3/2 ... 2/3
8/1 ... 5/2 ... 5/3 ... 3/4
9/1 ... 7/2 ... 8/3 .... 7/4 .. 4/5
10/1 .. 9/2 ... 11/3 .. 11/4 .. 9/5 ... 5/6
11/1 .. 11/2 .. 14/3 .. 15/4 .. 14/5 .. 11/6 .. 6/7 .. 1/3
The numerators, by rows: 1,2,3,4,5,6,1,7,3,2,8,5,5,3,9,7,8,7,4,10,9,...
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MATHEMATICA
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z = 23; g[1] = {1}; g[2] = {2}; g[3] = {3}; g[4] = {4}; g[5] = {5};
g[n_] := Reverse[Union[1 + g[n - 1], g[n - 5]/(1 + g[n - 5])]]
Table[g[n], {n, 1, 9}]
v = Flatten[Table[g[n], {n, 1, z}]];
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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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