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A324796 Irregular triangle read by rows: row n gives numerators of fractions in the Farey subsequence B(m).

%I A324796 #19 Jun 25 2022 09:38:33
%S A324796 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,
%T A324796 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,
%U A324796 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
%N A324796 Irregular triangle read by rows: row n gives numerators of fractions in the Farey subsequence B(m).
%C A324796 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.
%C A324796 A049691 gives the row lengths.
%D A324796 A. O. Matveev, Farey Sequences, De Gruyter, 2017.
%H A324796 A. O. Matveev, <a href="https://github.com/andreyomatveev/farey-sequences">Farey Sequences: Errata + Haskell code</a>
%e A324796 The first few sequences B(1), B(2), B(3), B(4) are:
%e A324796 [0, 1/2, 1],
%e A324796 [0, 1/3, 1/2, 2/3, 1],
%e A324796 [0, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1],
%e A324796 [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],
%e A324796 ...
%p A324796 Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
%p A324796 B := proc(m) local a,i,h,k; global Farey; a:=[];
%p A324796 for i in Farey(2*m) do
%p A324796 h:=numer(i); k:=denom(i);
%p A324796 if (h <= m) and (k-m <= h) then a:=[op(a),i]; fi; od: a; end;
%Y A324796 Cf. A006842/A006843, A049691, A324797 (denominators).
%K A324796 nonn,frac,tabf
%O A324796 1,7
%A A324796 _N. J. A. Sloane_, Sep 10 2019