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A006842 Triangle read by rows: row n gives numerators of Farey series of order n.
(Formerly M0041)
25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964

Bruckheimer, Maxim, and Abraham Arcavi. "Farey series and Pick’s area theorem." The mathematical intelligencer 17.4 (1995): 64-67.

Cobeli, Cristian, and Alexandru Zaharescu. "The Haros-Farey sequence at two hundred years." Acta Univ. Apulensis Math. Inform 5 (2003): 1-38.

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 Fractions, 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

Andrey O. Matveev, Neighboring Fractions in Farey Subsequences, arXiv:0801.1981 [math.NT], 2008-2010.

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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

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

Sequence in context: A205217 A054635 A003137 * A299038 A273693 A219967

Adjacent sequences:  A006839 A006840 A006841 * A006843 A006844 A006845

KEYWORD

nonn,nice,frac,tabf

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified October 23 03:21 EDT 2019. Contains 328335 sequences. (Running on oeis4.)