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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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Srinivasa Ramanujan, Squaring the circle, Wikisource, Journal of the Indian Mathematical Society, v, 1913, page 132.
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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
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CROSSREFS
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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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STATUS
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approved
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