A001203 Simple continued fraction expansion of Pi.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• cofr

Comments

The first 5821569425 terms were computed by Eric W. Weisstein on Sep 18 2011.

The first 10672905501 terms were computed by Eric W. Weisstein on Jul 17 2013.

The first 15000000000 terms were computed by Eric W. Weisstein on Jul 27 2013.

The first 30113021586 terms were computed by Syed Fahad on Apr 27 2021.

Examples

			Pi = 3.1415926535897932384...
   = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))
   = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 1, 292, ...].
		

References

• P. Beckmann, "A History of Pi".
• C. Brezinski, History of Continued Fractions and Padé Approximants, Springer-Verlag, 1991; pp. 151-152.
• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
• J. R. Goldman, The Queen of Mathematics, 1998, p. 50.
• R. S. Lehman, A Study of Regular Continued Fractions. Report 1066, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, Feb 1959.
• G. Lochs, Die ersten 968 Kettenbruchnenner von Pi. Monatsh. Math. 67 1963 311-316.
• C. D. Olds, Continued Fractions, Random House, NY, 1963; front cover of paperback edition.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 274.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..19999 [from the Plouffe web page]
• James Barton, Simple Continued Fraction Expansion of Pi [From _Lekraj Beedassy_, Oct 27 2008]
• E. Bombieri and A. J. van der Poorten, Continued fractions of algebraic numbers
• K. Y. Choong, D. E. Daykin and C. R. Rathbone, Regular continued fractions for pi and gamma, Math. Comp., 25 (1971), 403.
• Sebastian M. Cioabă and Werner Linde, A Bridge to Advanced Mathematics: from Natural to Complex Numbers, Amer. Math. Soc. (2023) Vol. 58, see page 360.
• Francesco Dolce and Pierre-Adrien Tahay, Column representation of Sturmian words in cellular automata, Czech Technical University (Prague, Czechia, 2022).
• Eduardo Dorrego López and Elías Fuentes Guillén, An Annotated Translation of Lambert's Vorläufige Kenntnisse (1766/1770), In: Irrationality, Transcendence and the Circle-Squaring Problem. Logic, Epistemology, and the Unity of Science (LEUS 2023) Springer, Cham. Vol 58.
• Exploratorium, 180 million terms of the simple CFE of pi
• Syed Fahad, 30 billion terms of the simple continued fraction of Pi
• Bill Gosper, answer to: Did Gosper or the Borweins first prove Ramanujans formula?, History of Science and Mathematics Stack Exchange, April 2020.
• Bill Gosper and Julian Ziegler Hunts, Animation
• B. Gourevitch, L'univers de Pi
• Hans Havermann, Simple Continued Fraction for Pi [a 483 MB file containing 180 million terms]
• Hans Havermann, Binary plot of 2^10 terms
• Maxim Sølund Kirsebom, Extreme Value Theory for Hurwitz Complex Continued Fractions, Entropy (2021) Vol. 23, No. 7, 840.
• Antony Lee, Diophantine Approximation and Dynamical Systems, Master's Thesis, Lund University (Sweden 2020).
• Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365 [math.QA], 2019.
• Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
• Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
• Simon Plouffe, 20 megaterms of this sequence as computed by Hans Havermann, starting in file CFPiTerms20aa.txt
• Denis Roegel, Lambert's proof of the irrationality of Pi: Context and translation, hal-02984214 [math.HO], 2020.
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
• Eric Weisstein's World of Mathematics, Pi Continued Fraction
• Eric Weisstein's World of Mathematics, Pi
• G. Xiao, Contfrac
• Index entries for continued fractions for constants
• Index entries for sequences related to the number Pi

Crossrefs

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

Cf. A097545, A097546.

Programs

Maple

cfrac (Pi,70,'quotients'); # Zerinvary Lajos, Feb 10 2007

Mathematica

ContinuedFraction[Pi, 98]

PARI

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

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

Python

import itertools as it; import sympy as sp
list(it.islice(sp.continued_fraction_iterator(sp.pi),100))
# Nicholas Stefan Georgescu, Feb 27 2023

Sage

continued_fraction(RealField(333)(pi)) # Peter Luschny, Feb 16 2015

Extensions

Word "Simple" added to the title by David Covert, Dec 06 2016