A006842 Triangle read by rows: row n gives numerators of Farey series of order n.

0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 1, 0, 1, 1, 1, 1, 2, 1, 2, 3, 1, 4, 3, 2, 5, 3, 4, 5, 6, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 5, 6, 7, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 4, 1, 5, 4, 3, 5, 2, 5, 3, 7, 4, 5, 6, 7, 8, 1

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• frac
• tabf

Examples

			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, 1/2, 2/3, 3/4, 1/1;
0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1;
... = A006842/A006843
		

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964
• J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 152.
• L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923. See Vol. 1.
• Guthery, Scott B. A motif of mathematics. Docent Press, 2011.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 23.
• W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
• A. O. Matveev, Farey Sequences, De Gruyter, 2017.
• I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 141.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10563
• Maxim Bruckheimer and Abraham Arcavi, Farey series and Pick’s area theorem, The Mathematical Intelligencer, 17.4 (1995): 64-67.
• Cristian Cobeli and Alexandru Zaharescu, The Haros-Farey sequence at two hundred years, Acta Univ. Apulensis Math. Inform 5 (2003): 1-38.
• Andrey O. Matveev, Neighboring Fractions in Farey Subsequences, arXiv:0801.1981 [math.NT], 2008-2010.
• Andrey O. Matveev, Farey Sequences: Errata + Haskell code
• N. J. A. Sloane, Stern-Brocot or Farey Tree
• Eric Weisstein's World of Mathematics, Farey Sequence.
• Index entries for sequences related to Stern's sequences

Crossrefs

Row n has A005728(n) terms. - Michel Marcus, Jun 27 2014

Cf. A006843 (denominators), A049455, A049456, A007305, A007306. Also A177405/A177407.

Programs

Maple

Farey := proc(n) sort(convert(`union`({0},{seq(seq(m/k,m=1..k),k=1..n)}),list)) end: seq(numer(Farey(i)),i=1..5); # Peter Luschny, Apr 28 2009

Mathematica

Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Flatten[ Table[ Numerator[ Farey[n]], {n, 0, 9}]] (* Robert G. Wilson v, Apr 08 2004 *)
Table[FareySequence[n] // Numerator, {n, 1, 9}] // Flatten (* Jean-François Alcover, Sep 25 2018 *)

PARI

row(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k););); vf = vecsort(Set(vf)); for (i=1, #vf, print1(numerator(vf[i]), ", "));} \\ Michel Marcus, Jun 27 2014

Extensions

More terms from Robert G. Wilson v, Apr 08 2004