A324797 Irregular triangle read by rows: row n gives denominators of fractions in the Farey subsequence B(m).

1, 2, 1, 1, 3, 2, 3, 1, 1, 4, 3, 5, 2, 5, 3, 4, 1, 1, 5, 4, 3, 5, 7, 2, 7, 5, 3, 4, 5, 1, 1, 6, 5, 4, 7, 3, 8, 5, 7, 9, 2, 9, 7, 5, 8, 3, 7, 4, 5, 6, 1, 1, 7, 6, 5, 4, 7, 3, 8, 5, 7, 9, 11, 2, 11, 9, 7, 5, 8, 3, 7, 4, 5, 6, 7, 1, 1, 8, 7, 6, 5, 9, 4, 7, 10, 3, 11, 8, 5, 12, 7, 9, 11, 13, 2, 13, 11, 9, 7, 12, 5, 8, 11, 3, 10, 7, 4, 9, 5, 6, 7, 8, 1

Offset

1,2

Comments

B(n) is denoted by F(B(2n),n) in Matveev (2017) - see definition on page 1. B(n) consists of the terms h/k of the Farey series F_{2n} such that k-n <= h <= n.

A049691 gives the row lengths.

References

A. O. Matveev, Farey Sequences, De Gruyter, 2017.

Links

Table of n, a(n) for n=1..113.

A. O. Matveev, Farey Sequences: Errata + Haskell code

Example

The first few sequences B(1), B(2), B(3), B(4) are:
[0, 1/2, 1],
[0, 1/3, 1/2, 2/3, 1],
[0, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1],
[0, 1/5, 1/4, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 3/4, 4/5, 1], [0, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 4/9, 1/2, 5/9, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 1],
...

Maple

Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
B := proc(m) local a, i, h, k; global Farey; a:=[];
for i in Farey(2*m) do
h:=numer(i); k:=denom(i);
if (h <= m) and (k-m <= h) then a:=[op(a), i]; fi; od: a; end;

Crossrefs

Cf. A006842/A006843, A049691, A324796 (numerators).

Sequence in context: A153901 A132844 A006843 * A049456 A117506 A179205

Adjacent sequences: A324794 A324795 A324796 * A324798 A324799 A324800

Keyword

nonn,frac,tabf

Author

N. J. A. Sloane, Sep 10 2019

Status

approved