%I A097545
%S A097545 1,0,1,2,3,4,7,10,13,16,19,22,25,47,69,91,113,135,157,179,201,223,245,
%T A097545 267,289,311,333,355,688,1043,1398,1753,2108,2463,2818,3173,3528,3883,
%U A097545 4238,4593,4948,5303,5658,6013,6368,6723,7078,7433,7788,8143,8498,8853
%N A097545 Numerators of "Farey fraction" approximations to Pi.
%C A097545 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:
%C A097545 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.
%C A097545 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 + ...))).
%D A097545 C. Brezinski, History of Continued Fractions and Pade' Approximants, 
               Springer-Verlag, 1991; pp. 151-152.
%D A097545 E. B. Christoffel, Observatio arithmetica, Ann. Math. Pura Appl., (II) 
               6 (1875), 148-153.
%H A097545 Dave Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/99/farey">
               Farey fractions on sci.math</a>
%e A097545 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, ...
%Y A097545 Cf. A097546.
%Y A097545 Sequence in context: A118426 A082008 A105330 this_sequence A073627 A062042 
               A107817
%Y A097545 Adjacent sequences: A097542 A097543 A097544 this_sequence A097546 A097547 
               A097548
%K A097545 nonn,frac,nice,easy
%O A097545 0,4
%A A097545 N. J. A. Sloane (njas(AT)research.att.com), Aug 28 2004
%E A097545 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), 
               May 08 2006