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A063673
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Denominators of convergents to Pi by Farey fractions.
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8
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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).
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EXAMPLE
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333/106 = 3.1415094... is 99.99% accurate;
355/113 = 3.1415929... is 99.99999% accurate.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A063674, A057082.
Sequence in context: A010754 A187807 A051036 * A105737 A335186 A257816
Adjacent sequences: A063670 A063671 A063672 * A063674 A063675 A063676
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KEYWORD
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frac,nonn
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AUTHOR
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Suren L. Fernando (fernando(AT)truman.edu), Jul 27 2001
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EXTENSIONS
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More terms from Charles R Greathouse IV, Aug 23 2006, and from M. F. Hasler, Apr 03 2013
Name simplified by Robert G. Wilson v, May 11 2020
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STATUS
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approved
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