

A324796


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


2



0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 0, 1, 1, 1, 2, 3, 1, 4, 3, 2, 3, 4, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 1, 5, 4, 3, 5, 2, 5, 3, 4, 5, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 1, 6, 5, 4, 3, 5, 2, 5, 3, 4, 5, 6, 1, 0, 1, 1, 1, 1, 2, 1, 2, 3, 1, 4, 3, 2, 5, 3, 4, 5, 6, 1, 7, 6, 5, 4, 7, 3, 5, 7, 2, 7, 5, 3, 7, 4, 5, 6, 7, 1
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OFFSET

1,7


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 kn <= h <= n.
A049691 gives the row lengths.


REFERENCES

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


LINKS

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


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 (km <= h) then a:=[op(a), i]; fi; od: a; end;


CROSSREFS

Cf. A006842/A006843, A049691, A324797 (denominators).
Sequence in context: A204172 A126304 A280522 * A049455 A322975 A133734
Adjacent sequences: A324793 A324794 A324795 * A324797 A324798 A324799


KEYWORD

nonn,frac,tabf


AUTHOR

N. J. A. Sloane, Sep 10 2019


STATUS

approved



