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A001203 Simple continued fraction expansion of Pi.
(Formerly M2646 N1054)
54
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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.
REFERENCES
P. Beckmann, "A History of Pi".
C. Brezinski, History of Continued Fractions and Padé Approximants, Springer-Verlag, 1991; pp. 151-152.
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).
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..19999 [from the Plouffe web page]
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.
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]
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
EXAMPLE
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, ...].
MAPLE
cfrac (Pi, 70, 'quotients'); # Zerinvary Lajos, Feb 10 2007
MATHEMATICA
ContinuedFraction[Pi, 98]
PROG
(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
(Sage) continued_fraction(RealField(333)(pi)) # Peter Luschny, Feb 16 2015
(Python)
import itertools as it; import sympy as sp
list(it.islice(sp.continued_fraction_iterator(sp.pi), 100))
CROSSREFS
Cf. A000796 for decimal expansion. See A007541 or A033089, A033090 for records.
Sequence in context: A106363 A128658 A234042 * A154883 A302029 A109732
KEYWORD
nonn,nice,cofr
AUTHOR
EXTENSIONS
Word "Simple" added to the title by David Covert, Dec 06 2016
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)