A063673 Denominators of convergents to Pi by Farey fractions.

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

Offset: 1

Author

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

Keywords

• frac
• nonn

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

Examples

			333/106 = 3.1415094... is 99.99% accurate;
355/113 = 3.1415929... is 99.99999% accurate.
		

Links

• P. D. Howard, Table of n, a(n) for n = 1..18865
• Ainsworth, Dawson, Piianta, and Warwick, The Farey Sequence.
• Krishnan Balasubramanian and Ernest R. Davidson, Rational approximations to pie: transcendental pi and Euler's Constant e, J. Math. Chem. (2023).
• Bhavsar and Thaker, Rational Approximation 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 Sequence.
• Dylan Zukin, The Farey Sequence and Its Niche(s).

Crossrefs

Cf. A063674, A057082.

Programs

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 *)

PARI

A063673(limit)= my(best=Pi-3, tmp); for(n=1,limit, tmp=abs(round(Pi*n)/n-Pi); if(tmpCharles R Greathouse IV, Aug 23 2006
(APL (NARS2000)) B⍸∪⌊\B←|(○1)-(⌊.5+○A)÷A←⍳100000 \\ Michael Turniansky, Jun 09 2015

Extensions

More terms from Charles R Greathouse IV, Aug 23 2006

More terms from M. F. Hasler, Apr 03 2013

Name simplified by Robert G. Wilson v, May 11 2020