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A006842
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Triangle read by rows: row n gives numerators of Farey series of order n.
(Formerly M0041)
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20
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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
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OFFSET
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1,9
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REFERENCES
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J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 152
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.
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).
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LINKS
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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
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,nice,frac,tabf
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Robert G. Wilson v, Apr 08 2004
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STATUS
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approved
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