A049455 -id:A049455 - cp's OEIS frontend

A294446 The tree of Farey fractions (or the Stern-Brocot tree), read across rows (the fraction i/j is represented as the pair i,j).

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

Offset: 0

N. J. A. Sloane, Nov 21 2017

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.

			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]]
...

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).

Index entries for fraction trees

Cf. A007305, A007306, A049455, A049456.
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.

# 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:

A089595 Table T(n,k), n>=0 and k>=0: Stern's diatomic array read by antidiagonals (version 5).

Original entry on oeis.org

1, 0, 1, -1, 1, 1, -3, 0, 2, 1, -2, -1, 1, 3, 1, -7, -1, 1, 2, 4, 1, -5, -4, 0, 3, 3, 5, 1, -8, -3, -1, 1, 5, 4, 6, 1, -3, -5, -1, 2, 2, 7, 5, 7, 1, -13, -2, -2, 1, 5, 3, 9, 6, 8, 1, -10, -9, -1, 1, 3, 8, 4, 11, 7, 9, 1, -17, -7, -5, 0, 4, 5, 11, 5, 13, 8, 10, 1, -7, -12, -4, -1, 1, 7, 7, 14, 6, 15, 9, 11, 1, -18, -5, -7, -1, 3, 2, 10, 9, 17, 7

Offset: 0

Philippe Deléham, Dec 30 2003

			row n=0 : 1, 0, -1, -3, -2, -7, -5, -8, -3, -13, -10, -17, -7, -18, -11, ...
row n=1 : 1, 1, 0, -1, -1, -4, -3, -5, -2, -9, -7, -12, -5, -13, ...
row n=2 : 1, 2, 1, 1, 0, -1, -1, -2, -1, -5, -4, -7, -3, ...
row n=3 : 1, 3, 2, 3, 1, 2, 1, 1, 0, -1, -1, -2, -1, ...
row n=4 : 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, ...

Cf. A049456, A070878, A070879, A049455, A002487.

Each row is obtained by copying the previous row but interpolating the sum of pairs of adjacent terms.
T(n, 2*k) = T(n-1, k) = T(n, k) - A002487(k).
T(n, 2*k+1) = T(n, 2*k) + T(n, 2*k+2); T(0, 0)=1, T(0, 1)=0.
The k-th column is an arithmetic progression with : T(n, k) = T(0, k) + n* A002487(k).

cp's OEIS Frontend

A294446 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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