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.
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).
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<best, best=tmp; print1(n", "))) \\ Charles R Greathouse IV, Aug 23 2006
(APL (NARS2000)) B⍸∪⌊\B←|(○1)-(⌊.5+○A)÷A←⍳100000 \\ Michael Turniansky, Jun 09 2015
CROSSREFS
KEYWORD
frac,nonn
AUTHOR
Suren L. Fernando (fernando(AT)truman.edu), Jul 27 2001
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
STATUS
approved