Search: id:A097545 Results 1-1 of 1 results found. %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, Farey fractions on sci.math %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 Search completed in 0.001 seconds