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A001203 Simple continued fraction expansion of Pi.
(Formerly M2646 N1054)

%I M2646 N1054

%S 3,7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,2,1,1,15,3,13,1,4,2,

%T 6,6,99,1,2,2,6,3,5,1,1,6,8,1,7,1,2,3,7,1,2,1,1,12,1,1,1,3,1,1,8,1,1,

%U 2,1,6,1,1,5,2,2,3,1,2,4,4,16,1,161,45,1,22,1,2,2,1,4,1,2,24,1,2,1,3,1,2,1

%N Simple continued fraction expansion of Pi.

%C The first 5821569425 terms were computed by _Eric W. Weisstein_ on Sep 18 2011.

%C The first 10672905501 terms were computed by _Eric W. Weisstein_ on Jul 17 2013.

%C The first 15000000000 terms were computed by _Eric W. Weisstein_ on Jul 27 2013.

%D P. Beckmann, "A History of Pi".

%D C. Brezinski, History of Continued Fractions and Padé Approximants, Springer-Verlag, 1991; pp. 151-152.

%D J. R. Goldman, The Queen of Mathematics, 1998, p. 50.

%D R. S. Lehman, A Study of Regular Continued Fractions. Report 1066, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, Feb 1959.

%D G. Lochs, Die ersten 968 Kettenbruchnenner von Pi. Monatsh. Math. 67 1963 311-316.

%D C. D. Olds, Continued Fractions, Random House, NY, 1963; front cover of paperback edition.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H N. J. A. Sloane, <a href="/A001203/b001203.txt">Table of n, a(n) for n = 0..19999</a> [from the Plouffe web page]

%H James Barton, <a href="http://www.virtuescience.com/pi-in-other-bases.html">Simple Continued Fraction Expansion of Pi</a> [From _Lekraj Beedassy_, Oct 27 2008]

%H E. Bombieri and A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/a113.pdf">Continued fractions of algebraic numbers</a>

%H K. Y. Choong, D. E. Daykin and C. R. Rathbone, <a href="https://doi.org/10.1090/S0025-5718-71-99719-5">Regular continued fractions for pi and gamma</a>, Math. Comp., 25 (1971), 403.

%H Exploratorium, <a href="http://chesswanks.com/seq/cfpi/">180 million terms of the simple CFE of pi</a>

%H Bill Gosper, answer to: <a href="https://hsm.stackexchange.com/a/11620">Did Gosper or the Borweins first prove Ramanujans formula?</a>, StackExchange, April 2020.

%H Bill Gosper and Julian Ziegler Hunts, <a href="/A001203/a001203.gif">Animation</a>

%H B. Gourevitch, <a href="http://www.pi314.net">L'univers de Pi</a>

%H Hans Havermann, <a href="http://chesswanks.com/pxp/cfpi.html">Simple Continued Fraction for Pi</a> [a 483 MB file containing 180 million terms]

%H Hans Havermann, <a href="/A001203/a001203.png">Binary plot of 2^10 terms</a>

%H Antony Lee, <a href="http://lup.lub.lu.se/luur/download?func=downloadFile&amp;recordOId=9006930&amp;fileOId=9006936">Diophantine Approximation and Dynamical Systems</a>, Master's Thesis, Lund University (Sweden 2020).

%H Sophie Morier-Genoud, Valentin Ovsienko, <a href="https://arxiv.org/abs/1908.04365">On q-deformed real numbers</a>, arXiv:1908.04365 [math.QA], 2019.

%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.

%H Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/">20 megaterms of this sequence as computed by Hans Havermann</a>, starting in file CFPiTerms20aa.txt

%H Denis Roegel, <a href="https://hal.archives-ouvertes.fr/hal-02984214">Lambert's proof of the irrationality of Pi: Context and translation</a>, hal-02984214 [math.HO], 2020.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PiContinuedFraction.html">Pi Continued Fraction</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pi.html">Pi</a>

%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>

%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>

%H <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>

%e Pi = 3.1415926535897932384...

%e = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))

%e = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 1, 292, ...]

%p cfrac (Pi,70,'quotients'); # _Zerinvary Lajos_, Feb 10 2007

%t ContinuedFraction[Pi, 98]

%o (PARI) contfrac(Pi) \\ contfracpnqn(%) is also useful!

%o (PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi); for (n=1, 20000, write("b001203.txt", n, " ", x[n])); } \\ _Harry J. Smith_, Apr 14 2009

%o (Sage) continued_fraction(RealField(333)(pi)) # _Peter Luschny_, Feb 16 2015

%Y Cf. A000796 for decimal expansion. See A007541 or A033089, A033090 for records.

%Y Cf. A097545, A097546.

%K nonn,nice,cofr

%O 0,1

%A _N. J. A. Sloane_

%E Word "Simple" added to the title by _David Covert_, Dec 06 2016

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Last modified April 19 21:57 EDT 2021. Contains 343117 sequences. (Running on oeis4.)