%I #80 Jun 28 2023 10:53:46

%S 1,4,5,6,7,57,64,71,78,85,92,99,106,113,16604,16717,16830,16943,17056,

%T 17169,17282,17395,17508,17621,17734,17847,17960,18073,18186,18299,

%U 18412,18525,18638,18751,18864,18977,19090,19203,19316,19429,19542,19655,19768,19881,19994,20107,20220,20333,20446,20559,20672,20785

%N Denominators of convergents to Pi by Farey fractions.

%C 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.

%C 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

%C 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

%C 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

%C The sequence uses Farey fractions instead of continued fractions. - _Robert G. Wilson v_, May 10 2020

%H P. D. Howard, <a href="/A063673/b063673.txt">Table of n, a(n) for n = 1..18865</a>

%H Ainsworth, Dawson, Piianta, and Warwick, <a href="https://www.maths.ed.ac.uk/~v1ranick/fareyproject.pdf">The Farey Sequence</a>.

%H Krishnan Balasubramanian and Ernest R. Davidson, <a href="https://doi.org/10.1007/s10910-023-01480-w">Rational approximations to pie: transcendental pi and Euler's Constant e</a>, J. Math. Chem. (2023).

%H Bhavsar and Thaker, <a href="http://dynamicpublisher.org/gallery/16-ijsrr-d1511.f.pdf">Rational Approximation Using Farey Sequence: Review</a>.

%H Das, Halder, Pratihar, and Bhowmick, <a href="https://arxiv.org/abs/1509.07757">Properties of Farey Sequence and their Applications to Digital Image Processing</a>, arXiv:1509.07757 [cs.OH], 2015.

%H Srinivasa Ramanujan, <a href="https://en.wikisource.org/wiki/Squaring_the_circle">Squaring the circle</a>, Wikisource, Journal of the Indian Mathematical Society, v, 1913, page 132.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FareySequence.html">Farey Sequence</a>.

%H Dylan Zukin, <a href="https://www.whitman.edu/Documents/Academics/Mathematics/2016/Zukin.pdf">The Farey Sequence and Its Niche(s)</a>.

%e 333/106 = 3.1415094... is 99.99% accurate;

%e 355/113 = 3.1415929... is 99.99999% accurate.

%t 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 *)

%o (PARI) A063673(limit)= my(best=Pi-3, tmp); for(n=1,limit, tmp=abs(round(Pi*n)/n-Pi); if(tmp<best, best=tmp; print1(n","))) \\ _Charles R Greathouse IV_, Aug 23 2006

%o (APL (NARS2000)) B⍸∪⌊\B←|(○1)-(⌊.5+○A)÷A←⍳100000 \\ _Michael Turniansky_, Jun 09 2015

%Y Cf. A063674, A057082.

%K frac,nonn

%O 1,2

%A Suren L. Fernando (fernando(AT)truman.edu), Jul 27 2001

%E More terms from _Charles R Greathouse IV_, Aug 23 2006

%E More terms from _M. F. Hasler_, Apr 03 2013

%E Name simplified by _Robert G. Wilson v_, May 11 2020