login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A097545 Numerators of "Farey fraction" approximations to Pi. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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 runs 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 + ...))).

REFERENCES

C. Brezinski, History of Continued Fractions and Pade' Approximants, Springer-Verlag, 1991; pp. 151-152.

E. B. Christoffel, Observatio arithmetica, Ann. Math. Pura Appl., (II) 6 (1875), 148-153.

LINKS

Table of n, a(n) for n=0..51.

Dave Rusin, Farey fractions on sci.math

EXAMPLE

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, ...

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-Fran├žois Alcover, May 18 2011 *)

CROSSREFS

Cf. A097546.

Sequence in context: A118426 A082008 A105330 * A202016 A073627 A062042

Adjacent sequences:  A097542 A097543 A097544 * A097546 A097547 A097548

KEYWORD

nonn,frac,nice,easy,look

AUTHOR

N. J. A. Sloane, Aug 28 2004

EXTENSIONS

More terms from Joshua Zucker, May 08 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 22 06:09 EST 2014. Contains 252328 sequences.