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A294446
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The tree of Farey fractions (or the Stern-Brocot tree), read across rows (the fraction i/j is represented as the pair i,j).
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2
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0, 1, 1, 2, 1, 1, 0, 1, 1, 3, 1, 2, 2, 3, 1, 1, 0, 1, 1, 4, 1, 3, 2, 5, 1, 2, 3, 5, 2, 3, 3, 4, 1, 1, 0, 1, 1, 5, 1, 4, 2, 7, 1, 3, 3, 8, 2, 5, 3, 7, 1, 2, 4, 7, 3, 5, 5, 8, 2, 3, 5, 7, 3, 4, 4, 5, 1, 1, 0, 1, 1, 6, 1, 5, 2, 9, 1, 4, 3, 11, 2, 7, 3, 10, 1, 3, 4, 11, 3, 8, 5, 13, 2, 5, 5, 12, 3, 7, 4, 9, 1, 2, 5, 9, 4
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OFFSET
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0,4
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COMMENTS
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The first row contains the fractions 0/1, 1/1,
and thereafter we copy the previous row, interpolating (a+c)/(b+d) between each pair of adjacent fractions a/b, c/d.
This version of the Farey tree contains the fractions in the range [0,1].
If we just look at the numerators we get A049455 and if we just look at the denominators we get A086596.
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REFERENCES
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W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
See A007305, A007306, A049455, A049456, etc. for many other references and links about the tree of Farey fractions (of which there are many versions).
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LINKS
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EXAMPLE
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This version of the tree begins as follows:
.................0/1..1/1
...............0/1..1/2..1/1
..........0/1..1/3..1/2..2/3..1/1
0/1..1/4..1/3..2/5..1/2..3/5..2/3..3/4..1/1
...
With the fractions written as pairs, the first few rows are:
[[0, 1], [1, 1]],
[[0, 1], [1, 2], [1, 1]],
[[0, 1], [1, 3], [1, 2], [2, 3], [1, 1]],
[[0, 1], [1, 4], [1, 3], [2, 5], [1, 2], [3, 5], [2, 3], [3, 4], [1, 1]],
[[0, 1], [1, 5], [1, 4], [2, 7], [1, 3], [3, 8], [2, 5], [3, 7], [1, 2], [4, 7,], [3, 5], [5, 8], [2, 3], [5, 7], [3, 4], [4, 5], [1, 1]]
...
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MAPLE
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# S[n] is the list of fractions, written as pairs [i, j], in row n of the triangle of Farey fractions
S[0]:=[[0, 1], [1, 1]];
for n from 1 to 6 do
S[n]:=[[0, 1]];
for k from 1 to nops(S[n-1])-1 do
a:=S[n-1][k][1]+S[n-1][k+1][1];
b:=S[n-1][k][2]+S[n-1][k+1][2];
S[n]:=[op(S[n]), [a, b], S[n-1][k+1]];
od:
lprint(S[n]);
od:
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CROSSREFS
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See A294442 for Kepler's tree of fractions.
For the number of distinct numerators in row n, see A293165, and for the distinct denominators see A293160.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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