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 A063673 Denominators of convergents to Pi by Farey fractions. 8
 1, 4, 5, 6, 7, 57, 64, 71, 78, 85, 92, 99, 106, 113, 16604, 16717, 16830, 16943, 17056, 17169, 17282, 17395, 17508, 17621, 17734, 17847, 17960, 18073, 18186, 18299, 18412, 18525, 18638, 18751, 18864, 18977, 19090, 19203, 19316, 19429, 19542, 19655, 19768, 19881, 19994, 20107, 20220, 20333, 20446, 20559, 20672, 20785 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Previous name: Denominators of sequence {3/1, 13/4, 16/5, 19/6, 22/7, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106, ... } of approximations to Pi with increasing denominators, where each approximation is an improvement on its predecessors. Pi = 3.1415926... is an irrational number and can't be exactly represented by a fraction with rational numerator and denominators. The fraction 355/113 is so accurate that improves the approximation of Pi by five significant digits over the previous 333/106. To find a slightly more accurate approximation we have to go to 52163 / 16604. - Sergio Pimentel, Sep 13 2005 The approximations 22/7 and 355/113 were already known by Zu Chongzhi (5th century) and A. Metius, 1585. (Thanks to P. Curtz for this remark.) - M. F. Hasler, Apr 03 2013 The approximation 355/113 was used by S. Ramanujan in the paper "Squaring the circle" to give a geometrical construction of a square whose area is approximately equal to that of a circle. See links. - Juan Monterde, Jul 26 2013 The sequence uses Farey fractions instead of continued fractions. - Robert G. Wilson v, May 10 2020 LINKS P. D. Howard, Table of n, a(n) for n = 1..18865 Ainsworth, Dawson, Piianta, and Warwick, The Farey Sequence. Bhavsar and Thaker, Rational Appoximation Using Farey Sequence: Review. Das, Halder, Pratihar, and Bhowmick, Properties of Farey Sequence and their Applications to Digital Image Processing, arXiv:1509.07757 [cs.OH], 2015. Srinivasa Ramanujan, Squaring the circle, Wikisource, Journal of the Indian Mathematical Society, v, 1913, page 132. Eric Weisstein's World of Mathematics, Farey Fractions. Dylan Zukin, The Farey Sequence and Its Niche(s). EXAMPLE 333/106 = 3.1415094... is 99.99% accurate; 355/113 = 3.1415929... is 99.99999% accurate. MATHEMATICA FareyConvergence[x_, n_] := Block[{n1 = 0, d1 = n9 = d9 = 1, F = 0, fp = FractionalPart@ x, lst}, \$MaxExtraPrecision = Max[50, n + 10]; lst = If[2 fp > 1, {Ceiling@ x}, {Floor@ x}]; While[d1 + d9 < n, a1 = n1/d1; a9 = n9/d9; n0 = n1 + n9; d0 = d1 + d9; a0 = n0/d0; If[a0 < fp, a1 = a0; n1 = n0; d1 = d0, a9 = a0; n9 = n0; d9 = d0]; If[Abs[fp - F] > Abs[fp - a0], F = a0; AppendTo[lst, a0 + IntegerPart@ x]]]; lst]; Denominator@ FareyConvergence[Pi, 10^10] (* Robert G. Wilson v, May 11 2020 *) PROG (PARI) A063673(limit)= my(best=Pi-3, tmp); for(n=1, limit, tmp=abs(round(Pi*n)/n-Pi); if(tmp

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Last modified December 7 13:08 EST 2021. Contains 349581 sequences. (Running on oeis4.)