A097545 Numerators of "Farey fraction" approximations to Pi.

1, 0, 1, 2, 3, 4, 7, 10, 13, 16, 19, 22, 25, 47, 69, 91, 113, 135, 157, 179, 201, 223, 245, 267, 289, 311, 333, 355, 688, 1043, 1398, 1753, 2108, 2463, 2818, 3173, 3528, 3883, 4238, 4593, 4948, 5303, 5658, 6013, 6368, 6723, 7078, 7433, 7788, 8143, 8498, 8853

Offset: 0

Author

N. J. A. Sloane, Aug 28 2004

Keywords

• nonn
• frac
• nice
• easy
• look

Comments

Given a real number x >= 1 (here x = Pi), start with 1/0 and 0/1 and construct the sequence of fractions f_n = r_n/s_n such that:

f_{n+1} = (r_k + r_n)/(s_k + s_n) where k is the greatest integer < n such that f_k <= x <= f_n. Sequence gives values r_n.

Write a 0 if f_n <= x and a 1 if f_n > x. This gives (for x = Pi) the sequence 1, 0, 0, 0, 1, 1, 1, 1, 0 (7 times), 1 (15 times), 0, 1, ... Ignore the initial string 1, 0, 0, 0, which is always the same. Look at the run lengths of the remaining sequence, which are in this case L_1 = 4, L_2 = 7, L_3 = 15, L_4 = 1, L_5 = 292, etc. (A001203). Christoffel showed that x has the continued fraction representation (L_1 - 1) + 1/(L_2 + 1/(L_3 + 1/(L_4 + ...))).

Examples

			The fractions are 1/0, 0/1, 1/1, 2/1, 3/1, 4/1, 7/2, 10/3, 13/4, 16/5, 19/6, 22/7, 25/8, 47/15, ...
		

References

• C. Brezinski, History of Continued Fractions and Padé Approximants, Springer-Verlag, 1991; pp. 151-152.
• E. B. Christoffel, Observatio arithmetica, Ann. Math. Pura Appl., (II) 6 (1875), 148-153.

Links

• Dave Rusin, Farey fractions on sci.math [Broken link]
• Dave Rusin, Farey fractions on sci.math [Cached copy]

Crossrefs

Cf. A097546.

Programs

Mathematica

f[x_, n_] := (m = Floor[x]; f0 = {m, m+1/2, m+1};
r = ({a___, b_, c_, d___} /; b < x < c) :> {b, (Numerator[b] + Numerator[c]) / (Denominator[b] + Denominator[c]), c}; Join[{m, m+1}, NestList[# /. r &, f0, n-3][[All, 2]]]); Join[{1, 0, 1, 2}, f[Pi, 48]] // Numerator  (* Jean-François Alcover, May 18 2011 *)

Extensions

More terms from Joshua Zucker, May 08 2006