A033089 -id:A033089 - cp's OEIS frontend

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

N. J. A. Sloane

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.

			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, ...].

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.

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

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

Cf. A097545, A097546.

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

Mathematica
```
ContinuedFraction[Pi, 98]
```

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

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

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

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

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

A033090 Indices of incrementally largest terms in the continued fraction for Pi.

Original entry on oeis.org

1, 2, 3, 5, 308, 432, 28422, 156382, 267314, 453294, 11504931, 849955263, 2349980289, 3588031780, 8600404591, 15621034283

Offset: 1

Eric W. Weisstein, Bill Gosper

This sequence assumes nonstandard indexing of continued fraction terms as [a_1; a_2, a_3, ...]. If you use the actual offset from A001203, corresponding to [a_0; a_1, a_2, ...], you get instead 0, 1, 2, 4, 307, 431, 28421, ... Compare with A033092 versus A224849. - Jeppe Stig Nielsen, Dec 14 2019

Syed Fahad, 30 billion terms of the simple continued fraction of Pi
E. Fontich, C. Simó, and A. Vieiro, On the "hidden" harmonics associated to best approximants due to quasiperiodicity in splitting phenomena, Regular and Chaotic Dynamics (2018), Pleiades Publishing, Vol. 23, Issue 6, 638-653. Also PDF.
Eric Weisstein's World of Mathematics, Pi
Eric Weisstein's World of Mathematics, Pi Continued Fraction

Cf. A033089, A001203.

With[{s = ContinuedFraction[Pi, 2*10^7]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Jan 31 2020 *)

a(12) from Eric W. Weisstein, Dec 08 2010
a(13) from Eric W. Weisstein, Sep 16 2011
a(14) from Eric W. Weisstein, Sep 17 2011
a(15) from Eric W. Weisstein, Jul 18 2013
a(16) from Syed Fahad, Apr 27 2021

A154883 Distinct entries in continued fraction for Pi in the order of their appearance.

Original entry on oeis.org

3, 7, 15, 1, 292, 2, 14, 84, 13, 4, 6, 99, 5, 8, 12, 16, 161, 45, 22, 24, 10, 26, 42, 9, 57, 18, 19, 30, 28, 20, 120, 23, 21, 127, 29, 11, 48, 436, 58, 34, 44, 20776, 94, 55, 32, 50, 43, 72, 33, 27, 36, 106, 17, 141, 39, 125, 41, 37, 25, 47, 61, 376, 107, 31

Offset: 1

Lee Corbin (lcorbin(AT)rawbw.com), Jan 16 2009

This is presumably a permutation of the positive integers. The inverse permutation (or "index sequence") A322778 begins 4,6,1,10,13,11,2,14,... and gives the position in the continued fraction of Pi at which 1, 2, 3, 4, 5, 6, ... first appear. - Remark corrected by N. J. A. Sloane, Jan 04 2019

The name means that when a number not yet in this sequence appears in the continued fraction of Pi, then that number is added to the sequence. - T. D. Noe, May 06 2013

			Since the actual continued fraction for Pi is 3, 7, 15, 1, 292, 1, 1, 1, 2, ..., this sequence begins 3, 7, 15, 1, 292, 2, ...

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000, May 06 2013

Cf. A001203, A033089 (for records of main continued fraction), A322778 (inverse), A033090.

DeleteDuplicates[ContinuedFraction[Pi,1000]] (* Harvey P. Dale, May 06 2013 *)
t = {}; s = ContinuedFraction[Pi, 1000]; Do[If[! MemberQ[t, s[[n]]], AppendTo[t, s[[n]]]], {n, Length[s]}]; t (* T. D. Noe, May 06 2013 *)

\p 10000
v=contfrac(Pi); for(i=1,#v,for(j=1,i-1,if(v[i]==v[j],v[i]=0;break))); v=select(n->n,v) \\ Charles R Greathouse IV, May 06 2013

More terms from Harvey P. Dale, May 05 2013

A007541 Incrementally largest terms in the continued fraction for Pi-2 (cf. A001203).

Original entry on oeis.org

1, 7, 15, 292, 436, 20776, 78629, 179136, 528210, 12996958, 878783625, 5408240597, 5916686112, 9448623833, 9787547328, 52662113289

Offset: 1

N. J. A. Sloane

nonn

No larger term in the first 10,672,905,501 terms of the c.f. - Eric W. Weisstein, Jul 18 2013

R. W. Gosper, Jr., Table of the simple continued fraction for pi and the derived decimal approximation, Math. Comp., 31 (1977), 1044.
R. W. Gosper, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
See A001203 for many additional references and links.

H. Havermann, Simple Continued Fraction for Pi [a 483 MB file containing 180 million terms]
Eric Weisstein's World of Mathematics, Pi
Eric Weisstein's World of Mathematics, Pi Continued Fraction
Index entries for sequences related to the number Pi

Cf. A001203, A000796, A033090.

Apart from initial term, same as A033089.

upto=10^7;a={};r=0;f=ContinuedFraction[Pi-2,upto];Do[If[f[[i]]>r,AppendTo[a,r=f[[i]]]],{i, upto}];a (* Paolo Xausa, Nov 28 2021 *)

allocatemem(4096*10^6);
default(realprecision, 50000);
v = contfrac(Pi-2);
m = 0;
for (i=1, #v, if (v[i] > m, m = v[i]; print1(m, ", "))); \\ Michel Marcus, Aug 05 2017; to get 7 terms

Corrected (missing a(9) added) by Eric W. Weisstein, Dec 08 2010
a(12) from Eric W. Weisstein, Dec 08 2010
a(13) from Eric W. Weisstein, Sep 16 2011
a(14) from Eric W. Weisstein, Sep 17 2011
a(15) from Eric W. Weisstein, Jul 18 2013
a(6) corrected by Bobby Jacobs, Aug 05 2017
a(16) = A033089(16) from Jeppe Stig Nielsen, Nov 28 2021

A203168 Positions of 1 in the continued fraction expansion of Pi.

Original entry on oeis.org

4, 6, 7, 8, 10, 12, 15, 16, 21, 24, 25, 29, 35, 41, 42, 45, 47, 51, 53, 54, 56, 57, 58, 60, 61, 63, 64, 66, 68, 69, 74, 79, 82, 84, 87, 89, 92, 94, 96, 98, 99, 104, 108, 113, 115, 116, 121, 125, 126, 134, 136, 138, 141, 144, 148, 149, 150, 154, 157, 158, 160

Offset: 1

Ben Branman, Dec 29 2011

In the Gauss-Kuzmin distribution, 1 appears with probability log_2(4/3) = 41.5037...%. Thus the n-th appearance of 1 in the continued fraction of a real number chosen uniformly from [0, 1) will be, with probability 1, n / (log_2(4/3)) + O(sqrt(n)). Does this sequence have the same asymptotic? - Charles R Greathouse IV, Dec 30 2011

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pi Continued Fraction
Index entries for continued fractions for constants
Index entries for sequences related to the number Pi

Cf. A001203, A033089, A000796.

Flatten[Position[ContinuedFraction[Pi, 160], 1]]

v=contfrac(Pi);for(i=1,#v,if(v[i]==1,print1(i", "))) \\ Charles R Greathouse IV, Dec 30 2011

A001203(a(n)) = 1.

cp's OEIS Frontend

A001203 Simple continued fraction expansion of Pi.

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A033090 Indices of incrementally largest terms in the continued fraction for Pi.

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A154883 Distinct entries in continued fraction for Pi in the order of their appearance.

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A007541 Incrementally largest terms in the continued fraction for Pi-2 (cf. A001203).

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A203168 Positions of 1 in the continued fraction expansion of Pi.

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