This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000796 M2218 N0880 #761 Aug 07 2025 07:58:00
%S A000796 3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,
%T A000796 8,8,4,1,9,7,1,6,9,3,9,9,3,7,5,1,0,5,8,2,0,9,7,4,9,4,4,5,9,2,3,0,7,8,
%U A000796 1,6,4,0,6,2,8,6,2,0,8,9,9,8,6,2,8,0,3,4,8,2,5,3,4,2,1,1,7,0,6,7,9,8,2,1,4
%N A000796 Decimal expansion of Pi (or digits of Pi).
%C A000796 Sometimes called Archimedes's constant.
%C A000796 Ratio of a circle's circumference to its diameter.
%C A000796 Also area of a circle with radius 1.
%C A000796 Also surface area of a sphere with diameter 1.
%C A000796 A useful mnemonic for remembering the first few terms: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics ...
%C A000796 Also ratio of surface area of sphere to one of the faces of the circumscribed cube. Also ratio of volume of a sphere to one of the six inscribed pyramids in the circumscribed cube. - _Omar E. Pol_, Aug 09 2012
%C A000796 Also surface area of a quarter of a sphere of radius 1. - _Omar E. Pol_, Oct 03 2013
%C A000796 Also the area under the peak-shaped even function f(x)=1/cosh(x). Proof: for the upper half of the integral, write f(x) = (2*exp(-x))/(1+exp(-2x)) = 2*Sum_{k>=0} (-1)^k*exp(-(2k+1)*x) and integrate term by term from zero to infinity. The result is twice the Gregory series for Pi/4. - _Stanislav Sykora_, Oct 31 2013
%C A000796 A curiosity: a 144 X 144 magic square of 7th powers was recently constructed by Toshihiro Shirakawa. The magic sum = 3141592653589793238462643383279502884197169399375105, which is the concatenation of the first 52 digits of Pi. See the MultiMagic Squares link for details. - Christian Boyer, Dec 13 2013 [Comment revised by _N. J. A. Sloane_, Aug 27 2014]
%C A000796 x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - _Omar E. Pol_, Dec 25 2013
%C A000796 Also diameter of a sphere whose surface area equals the volume of the circumscribed cube. - _Omar E. Pol_, Jan 13 2014
%C A000796 From _Daniel Forgues_, Mar 20 2015: (Start)
%C A000796 An interesting anecdote about the base-10 representation of Pi, with 3 (integer part) as first (index 1) digit:
%C A000796 358 0
%C A000796 359 3
%C A000796 360 6
%C A000796 361 0
%C A000796 362 0
%C A000796 And the circle is customarily subdivided into 360 degrees (although Pi radians yields half the circle)...
%C A000796 (End)
%C A000796 Sometimes referred to as Archimedes's constant, because the Greek mathematician computed lower and upper bounds of Pi by drawing regular polygons inside and outside a circle. In Germany it was called the Ludolphian number until the early 20th century after the Dutch mathematician Ludolph van Ceulen (1540-1610), who calculated up to 35 digits of Pi in the late 16th century. - _Martin Renner_, Sep 07 2016
%C A000796 As of the beginning of 2019 more than 22 trillion decimal digits of Pi are known. See the Wikipedia article "Chronology of computation of Pi". - _Harvey P. Dale_, Jan 23 2019
%C A000796 On March 14, 2019, Emma Haruka Iwao announced the calculation of 31.4 trillion digits of Pi using Google Cloud's infrastructure. - _David Radcliffe_, Apr 10 2019
%C A000796 Also volume of three quarters of a sphere of radius 1. - _Omar E. Pol_, Aug 16 2019
%C A000796 On August 5, 2021, researchers from the University of Applied Sciences of the Grisons in Switzerland announced they had calculated 62.8 trillion digits. Guinness World Records has not verified this yet. - _Alonso del Arte_, Aug 23 2021
%C A000796 The Hermite-Lindemann (1882) theorem states, that if z is a nonzero algebraic number, then e^z is a transcendent number. The transcendence of Pi then results from Euler's relation: e^(i*Pi) = -1. - _Peter Luschny_, Jul 21 2023
%C A000796 The 10000 words of the book "Not A Wake" by Michael Keith, written in Pilish, match in length the first 10000 digits of Pi. - _Paolo Xausa_, Aug 07 2025
%D A000796 Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.
%D A000796 J. Arndt & C. Haenel, Pi Unleashed, Springer NY 2001.
%D A000796 P. Beckmann, A History of Pi, Golem Press, Boulder, CO, 1977.
%D A000796 Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 396.
%D A000796 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 237-239.
%D A000796 J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997.
%D A000796 P. Eyard and J.-P. Lafon, The Number Pi, Amer. Math. Soc., 2004.
%D A000796 S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.4.
%D A000796 Le Petit Archimede, Special Issue On Pi, Supplement to No. 64-5, May 1980 ADCS Amiens.
%D A000796 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 31.
%D A000796 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000796 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000796 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equations 1:7:1, 1:7:2 at pages 12-13.
%D A000796 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See pp. 48-55.
%H A000796 Harry J. Smith, <a href="/A000796/b000796.txt">Table of n, a(n) for n = 1..20000</a>
%H A000796 Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, and Brendan M. Quine, <a href="https://arxiv.org/abs/2004.11711">Unconditional applicability of the Lehmer's measure to the two-term Machin-like formula for pi</a>, arXiv:2004.11711 [math.GM], 2020.
%H A000796 Emilio Ambrisi and Bruno Rizzi, <a href="https://www.matmedia.it/wp-content/uploads/2024/07/Quaderno-Mondragone.pdf">Appunti da un corso di aggiornamento</a>, Mathesis (Sezione Casertana), Quaderno n. 1, Liceo G. Galilei, Mondragone (CE), Italy, June 22-28 1979. (In Italian). See p. 15.
%H A000796 Dave Andersen, <a href="http://www.angio.net/pi/piquery">Pi-Search Page</a>
%H A000796 Anonymous, <a href="http://web.archive.org/web/20140225153300/http://www.exploratorium.edu/pi/pi_archive/Pi10-6.html">A million digits of Pi</a>
%H A000796 Anonymous, <a href="http://mapage.noos.fr/echolalie/l127.htm">Liste de quelques milliers de decimales du nombre de pi</a>
%H A000796 D. H. Bailey, <a href="https://web.archive.org/web/20100826224951/http://www.nersc.gov:80/homes/dhbailey/dhbpapers/dhb-kanada.pdf">On Kanada's computation of 1.24 trillion digits of Pi</a> [archived page]
%H A000796 D. H. Bailey and J. M. Borwein, <a href="http://www.ams.org/notices/200505/fea-borwein.pdf">Experimental Mathematics: Examples, Methods and Implications</a>, Notices of the AMS, Volume 52, Number 5, May 2005, pp. 502-514.
%H A000796 Harry Baker, <a href="https://www.livescience.com/record-number-of-pi-digits.html">"Pi calculated to a record-breaking 62.8 trillion digits"</a>, Live Science, August 17, 2021.
%H A000796 Steve Baker and Thomas Moore, <a href="https://storage.googleapis.com/pi100t/index.html">100 trillion digits of pi</a>
%H A000796 Frits Beukers, <a href="http://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-372.pdf">A rational approach to Pi</a>, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.
%H A000796 J. M. Borwein, <a href="http://www.cecm.sfu.ca/~jborwein/pi_cover.html">Talking about Pi</a>
%H A000796 J. M. Borwein and M. Macklem, <a href="http://www.austms.org.au/Gazette/2006/Sep06/pi.pdf">The (Digital) Life of Pi</a>, The Australian Mathematical Society Gazette, Volume 33, Number 5, Sept. 2006, pp. 243-248.
%H A000796 Peter Borwein, <a href="http://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-3-254.pdf">The amazing number Pi</a>, Nieuw Archief voor de Wiskunde, September 2000, pp. 254-258.
%H A000796 Christian Boyer, <a href="http://www.multimagie.com/">MultiMagic Squares</a>
%H A000796 J. Britton, <a href="https://web.archive.org/web/20170701164231/http://britton.disted.camosun.bc.ca/jbpimem.htm">Mnemonics For The Number Pi</a> [archived page]
%H A000796 D. Castellanos, <a href="http://www.jstor.org/stable/2690037">The ubiquitous pi</a>, Math. Mag., 61 (1988), 67-98 and 148-163.
%H A000796 Jonas Castillo Toloza, <a href="http://www.lifesmith.com/mathfun.html#41">Fascinating Method for Finding Pi</a>
%H A000796 E. S. Croot, <a href="http://people.math.gatech.edu/~ecroot/transcend.pdf">Pade Approximations and the Transcendence of pi</a>
%H A000796 L. Euler, <a href="https://arxiv.org/abs/math/0506415">On the sums of series of reciprocals</a>, arXiv:math/0506415 [math.HO], 2005-2008.
%H A000796 L. Euler, <a href="http://eulerarchive.maa.org/pages/E041.html">De summis serierum reciprocarum</a>, E41.
%H A000796 Eureka, <a href="http://users.skynet.be/ekurea/toutpi.html">Tout pi or not tout pi</a>
%H A000796 Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/publist.html">Zeta function expansions of some classical constants</a>
%H A000796 Jeremy Gibbons, <a href="http://www.cs.ox.ac.uk/jeremy.gibbons/publications/spigot.pdf">Unbounded Spigot Algorithms for the Digits of Pi</a>
%H A000796 GJ, <a href="http://web.archive.org/web/20011214030954/http://gj.mit.edu/pi/digits/10million.txt">10 million digits of Pi</a>
%H A000796 X. Gourdon, <a href="https://web.archive.org/web/20160428024740/http://webs.adam.es:80/rllorens/pi.htm">Pi to 16000 decimals</a> [archived page]
%H A000796 Xavier Gourdon, <a href="http://numbers.computation.free.fr/Constants/Algorithms/nthdigit.html">A new algorithm for computing Pi in base 10</a>
%H A000796 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Pi/pi.html">Archimedes' constant Pi</a>
%H A000796 B. Gourevitch, <a href="http://www.pi314.net">L'univers de Pi</a>
%H A000796 Antonio Gracia Llorente, <a href="https://osf.io/preprints/osf/dg8tf">Novel Infinite Products πe and π/e</a>, OSF Preprint, 2024.
%H A000796 L. Grebelius, <a href="http://web.archive.org/web/20130303114650/http://www2.tripnet.se/~nlg/pi0001.htm">Approximation of Pi: First 1000000 digits</a>
%H A000796 J. Guillera and J. Sondow, <a href="http://dx.doi.org/10.1007/s11139-007-9102-0">Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent</a>, Ramanujan J. 16 (2008) 247-270. Preprint: <a href="https://arxiv.org/abs/math/0506319">arXiv:math/0506319</a> [math.NT] (2005-2006).
%H A000796 Carl-Johan Haster, <a href="https://arxiv.org/abs/2005.05472">Pi from the sky -- A null test of general relativity from a population of gravitational wave observations</a>, arXiv:2005.05472 [gr-qc], 2020.
%H A000796 H. Havermann, <a href="https://web.archive.org/web/20181130124011/http://chesswanks.com/pxp/cfpi.html">Simple Continued Fraction for Pi</a> [archived page]
%H A000796 M. D. Huberty et al., <a href="http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html">100000 Digits of Pi</a>
%H A000796 ICON Project, <a href="https://www2.cs.arizona.edu/icon/oddsends/pi.htm">Pi to 50000 places</a> [archived page]
%H A000796 Emma Haruka Iwao, <a href="https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud">Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes' constant on Google Cloud</a>
%H A000796 P. Johns, <a href="https://web.archive.org/web/20020207043745/http://www.wpdpi.com:80/pi.shtml">120000 Digits of Pi</a> [archived page]
%H A000796 Yasumasa Kanada, <a href="http://www.super-computing.org/">1.24 trillion digits of Pi</a>
%H A000796 Yasumasa Kanada and Daisuke Takahashi, <a href="https://web.archive.org/web/20050305084522/http://www.cecm.sfu.ca:80/personal/jborwein/Kanada_200b.html">206 billion digits of Pi</a> [archived page]
%H A000796 Literate Programs, <a href="https://web.archive.org/web/20150905214036/http://en.literateprograms.org/Pi_with_Machin%27s_formula_(Haskell)">Pi with Machin's formula (Haskell)</a> [archived page]
%H A000796 Johannes W. Meijer, <a href="/A000796/a000796.jpg">Pi everywhere</a> poster, Mar 14 2013
%H A000796 J. Moyer, <a href="http://www.rsok.com/~jrm/pi10000.txt">First 10000 digits of pi</a>
%H A000796 NERSC, <a href="http://pi.nersc.gov/">Search Pi</a> [broken link]
%H A000796 Remco Niemeijer, <a href="http://programmingpraxis.com/2009/02/20/the-digits-of-pi/">The Digits of Pi</a>, programmingpraxis.
%H A000796 Steve Pagliarulo, <a href="https://web.archive.org/web/20160820022833/http://members.shaw.ca:80/francislyster/pi/pi.html">Stu's pi page</a> [archived page]
%H A000796 Chittaranjan Pardeshi, <a href="/A000796/a000796.pdf">BBP-Like formula for Pi in Golden Ratio Base Phi</a>
%H A000796 Michael Penn, <a href="https://www.youtube.com/watch?v=dzzbhfudx5M">A nice inverse tangent integral.</a>, YouTube video, 2020.
%H A000796 Michael Penn, <a href="https://www.youtube.com/watch?v=dFKbVTHK4tU">Pi is irrational (π∉ℚ)</a>, YouTube video, 2020.
%H A000796 I. Peterson, <a href="http://web.cs.ucla.edu/~klinger/mathland_3_11.html">A Passion for Pi</a>
%H A000796 G. M. Phillips, <a href="http://www.mcs.st-and.ac.uk/~gmp/gmpCON.html">Table of contents of "Pi: A source Book"</a>
%H A000796 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/pi10000.txt">10000 digits of Pi</a>
%H A000796 Simon Plouffe, <a href="https://arxiv.org/abs/2201.12601">A formula for the nth decimal digit or binary of Pi and powers of Pi</a>, arXiv:2201.12601 [math.NT], 2022.
%H A000796 D. Pothet, <a href="http://perso.wanadoo.fr/didier.pothet/pi.html">Chronologie du calcul des decimales de pi</a> [broken link]
%H A000796 M. Z. Rafat and D. Dobie, <a href="https://arxiv.org/abs/1901.06260">Throwing Pi at a wall</a>, arXiv:1901.06260 [physics.class-ph], 2020.
%H A000796 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper6/page1.htm">Modular equations and approximations to \pi</a>, Quart. J. Math. 45 (1914), 350-372.
%H A000796 H. Ricardo, <a href="http://www.maa.org/press/maa-reviews/the-number-pi">Review of "The Number Pi" by P. Eymard & J.-P. Lafon</a>
%H A000796 M. Ripa and G. Morelli, <a href="http://www.iqsociety.org/general/documents/Retro_analytical_Reasoning_IQ_tests_for_the_High_Range.pdf">Retro-analytical Reasoning IQ tests for the High Range</a>, 2013.
%H A000796 Grant Sanderson, <a href="https://www.youtube.com/watch?v=jsYwFizhncE">Why do colliding blocks compute pi?</a>, 3Blue1Brown video (2019).
%H A000796 Daniel B. Sedory, <a href="http://thestarman.pcministry.com/math/pi/index.html">The Pi Pages</a>
%H A000796 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See p. 180.
%H A000796 D. Shanks and J. W. Wrench, Jr., <a href="http://dx.doi.org/10.1090/S0025-5718-1962-0136051-9">Calculation of pi to 100,000 decimals</a>, Math. Comp. 16 1962 76-99.
%H A000796 Jean-Louis Sigrist, <a href="http://jlsigrist.com/pi.html">Les 128000 premieres decimales du nombre PI</a>
%H A000796 Sizes, <a href="http://www.sizes.com/numbers/pi.htm">pi</a>
%H A000796 N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 5.
%H A000796 Katie Steckles, <a href="https://aperiodical.com/2015/03/pi-and-constrained-writing/">Pi and constrained writing</a>, The Aperiodical, 2015.
%H A000796 A. Sofo, <a href="http://www.emis.de/journals/JIPAM/images/084_05_JIPAM/084_05.pdf">Pi and some other constants</a>, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, Issue 5, Article 138, 2005.
%H A000796 Jonathan Sondow, <a href="https://arxiv.org/abs/math/0401406">A faster product for Pi and a new integral for ln Pi/2</a>, arXiv:math/0401406 [math.NT], 2004; Amer. Math. Monthly 112 (2005) 729-734.
%H A000796 D. Surendran, <a href="https://web.archive.org/web/20100128225616/http://www.uz.ac.zw:80/science/maths/zimaths/pimnem.htm">Can I have a small container of coffee?</a> [archived page]
%H A000796 Wislawa Szymborska, <a href="http://katherinestange.com/mathweb/p_p2.html">Pi (The admirable number Pi)</a>, Miracle Fair, 2002.
%H A000796 G. Vacca, <a href="http://dx.doi.org/10.1090/S0002-9904-1910-01919-4">A new analytical expression for the number pi, and some historical considerations</a>, Bull. Amer. Math. Soc. 16 (1910), 368-369.
%H A000796 Stan Wagon, <a href="http://pi314.at/math/normal.html">Is Pi Normal?</a>
%H A000796 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pi.html">Pi</a> and <a href="https://mathworld.wolfram.com/PiDigits.html">Pi Digits</a>
%H A000796 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula">Bailey-Borwein-Plouffe formula</a>, <a href="https://en.wikipedia.org/wiki/Normal_number">Normal Number</a>, <a href="https://www.wikipedia.org/wiki/Pi">Pi</a>, and <a href="https://en.wikipedia.org/wiki/Machin-like_formula">Machin-like formula</a>
%H A000796 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pilish">Pilish</a>.
%H A000796 Alexander J. Yee & Shigeru Kondo, <a href="http://www.numberworld.org/misc_runs/pi-5t/details.html">5 Trillion Digits of Pi - New World Record</a>
%H A000796 Alexander J. Yee & Shigeru Kondo, <a href="http://www.numberworld.org/misc_runs/pi-10t/details.html">Round 2... 10 Trillion Digits of Pi</a>
%H A000796 <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>
%H A000796 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000796 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A000796 Pi = 4*Sum_{k>=0} (-1)^k/(2k+1) [Madhava-Gregory-Leibniz, 1450-1671]. - _N. J. A. Sloane_, Feb 27 2013
%F A000796 From _Johannes W. Meijer_, Mar 10 2013: (Start)
%F A000796 2/Pi = (sqrt(2)/2) * (sqrt(2 + sqrt(2))/2) * (sqrt(2 + sqrt(2 + sqrt(2)))/2) * ... [Viete, 1593]
%F A000796 2/Pi = Product_{k>=1} (4*k^2-1)/(4*k^2). [Wallis, 1655]
%F A000796 Pi = 3*sqrt(3)/4 + 24*(1/12 - Sum_{n>=2} (2*n-2)!/((n-1)!^2*(2*n-3)*(2*n+1)*2^(4*n-2))). [Newton, 1666]
%F A000796 Pi/4 = 4*arctan(1/5) - arctan(1/239). [Machin, 1706]
%F A000796 Pi^2/6 = 3*Sum_{n>=1} 1/(n^2*binomial(2*n,n)). [Euler, 1748]
%F A000796 1/Pi = (2*sqrt(2)/9801) * Sum_{n>=0} (4*n)!*(1103+26390*n)/((n!)^4*396^(4*n)). [Ramanujan, 1914]
%F A000796 1/Pi = 12*Sum_{n>=0} (-1)^n*(6*n)!*(13591409 + 545140134*n)/((3*n)!*(n!)^3*(640320^3)^(n+1/2)). [David and Gregory Chudnovsky, 1989]
%F A000796 Pi = Sum_{n>=0} (1/16^n) * (4/(8*n+1) - 2/(8*n+4) - 1/(8*n+5) - 1/(8*n+6)). [Bailey-Borwein-Plouffe, 1989] (End)
%F A000796 Pi = 4 * Sum_{k>=0} 1/(4*k+1) - 1/(4*k+3). - _Alexander R. Povolotsky_, Dec 25 2008
%F A000796 Pi = 4*sqrt(-1*(Sum_{n>=0} (i^(2*n+1))/(2*n+1))^2). - _Alexander R. Povolotsky_, Jan 25 2009
%F A000796 Pi = Integral_{x=-oo..oo} dx/(1+x^2). - _Mats Granvik_ and _Gary W. Adamson_, Sep 23 2012
%F A000796 Pi - 2 = 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 - 1/36 + 1/45 + ... [Jonas Castillo Toloza, 2007], that is, Pi - 2 = Sum_{n>=1} (1/((-1)^floor((n-1)/2)*(n^2+n)/2)). - _José de Jesús Camacho Medina_, Jan 20 2014
%F A000796 Pi = 3 * Product_{t=img(r),r=(1/2+i*t) root of zeta function} (9+4*t^2)/(1+4*t^2) <=> RH is true. - _Dimitris Valianatos_, May 05 2016
%F A000796 From _Ilya Gutkovskiy_, Aug 07 2016: (Start)
%F A000796 Pi = Sum_{k>=1} (3^k - 1)*zeta(k+1)/4^k.
%F A000796 Pi = 2*Product_{k>=2} sec(Pi/2^k).
%F A000796 Pi = 2*Integral_{x>=0} sin(x)/x dx. (End)
%F A000796 Pi = 2^{k + 1}*arctan(sqrt(2 - a_{k - 1})/a_k) at k >= 2, where a_k = sqrt(2 + a_{k - 1}) and a_1 = sqrt(2). - _Sanjar Abrarov_, Feb 07 2017
%F A000796 Pi = Integral_{x = 0..2} sqrt(x/(2 - x)) dx. - _Arkadiusz Wesolowski_, Nov 20 2017
%F A000796 Pi = lim_{n->oo} 2/n * Sum_{m=1,n} ( sqrt( (n+1)^2 - m^2 ) - sqrt( n^2 - m^2 ) ). - _Dimitri Papadopoulos_, May 31 2019
%F A000796 From _Peter Bala_, Oct 29 2019: (Start)
%F A000796 Pi = Sum_{n >= 0} 2^(n+1)/( binomial(2*n,n)*(2*n + 1) ) - Euler.
%F A000796 More generally, Pi = (4^x)*x!/(2*x)! * Sum_{n >= 0} 2^(n+1)*(n+x)!*(n+2*x)!/(2*n+2*x+1)! = 2*4^x*x!^2/(2*x+1)! * hypergeom([2*x+1,1], [x+3/2], 1/2), valid for complex x not in {-1,-3/2,-2,-5/2,...}. Here, x! is shorthand notation for the function Gamma(x+1). This identity may be proved using Gauss's second summation theorem.
%F A000796 Setting x = 3/4 and x = -1/4 (resp. x = 1/4 and x = -3/4) in the above identity leads to series representations for the constant A085565 (resp. A076390). (End)
%F A000796 Pi = Im(log(-i^i)) = log(i^i)*(-2). - _Peter Luschny_, Oct 29 2019
%F A000796 From _Amiram Eldar_, Aug 15 2020: (Start)
%F A000796 Equals 2 + Integral_{x=0..1} arccos(x)^2 dx.
%F A000796 Equals Integral_{x=0..oo} log(1 + 1/x^2) dx.
%F A000796 Equals Integral_{x=0..oo} log(1 + x^2)/x^2 dx.
%F A000796 Equals Integral_{x=-oo..oo} exp(x/2)/(exp(x) + 1) dx. (End)
%F A000796 Equals 4*(1/2)!^2 = 4*Gamma(3/2)^2. - _Gary W. Adamson_, Aug 23 2021
%F A000796 From _Peter Bala_, Dec 08 2021: (Start)
%F A000796 Pi = 32*Sum_{n >= 1} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9))= 384*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)*(4*n^2 - 9)*(4*n^2 - 25)).
%F A000796 More generally, it appears that for k = 1,2,3,..., Pi = 16*(2*k)!*Sum_{n >= 1} (-1)^(n+k+1)*n^2/((4*n^2 - 1)* ... *(4*n^2 - (2*k+1)^2)).
%F A000796 Pi = 32*Sum_{n >= 1} (-1)^(n+1)*n^2/(4*n^2 - 1)^2 = 768*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)^2*(4*n^2 - 9)^2).
%F A000796 More generally, it appears that for k = 0,1,2,..., Pi = 16*Catalan(k)*(2*k)!*(2*k+2)!*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)^2* ... *(4*n^2 - (2*k+1)^2)^2).
%F A000796 Pi = (2^8)*Sum_{n >= 1} (-1)^(n+1)*n^2/(4*n^2 - 1)^4 = (2^17)*(3^5)*Sum_{n >= 2} (-1)^n*n^2*(n^2 - 1)/((4*n^2 - 1)^4*(4*n^2 - 9)^4) = (2^27)*(3^5)*(5^5)* Sum_{n >= 3} (-1)^(n+1)*n^2*(n^2 - 1)*(n^2 - 4)/((4*n^2 - 1)^4*(4*n^2 - 9)^4*(4*n^2 - 25)^4). (End)
%F A000796 For odd n, Pi = (2^(n-1)/A001818((n-1)/2))*gamma(n/2)^2. - _Alan Michael Gómez Calderón_, Mar 11 2022
%F A000796 Pi = 4/phi + Sum_{n >= 0} (1/phi^(12*n)) * ( 8/((12*n+3)*phi^3) + 4/((12*n+5)*phi^5) - 4/((12*n+7)*phi^7) - 8/((12*n+9)*phi^9) - 4/((12*n+11)*phi^11) + 4/((12*n+13)*phi^13) ) where phi = (1+sqrt(5))/2. - _Chittaranjan Pardeshi_, May 16 2022
%F A000796 Pi = sqrt(3)*(27*S - 36)/24, where S = A248682. - _Peter Luschny_, Jul 22 2022
%F A000796 Equals Integral_{x=0..1} 1/sqrt(x-x^2) dx. - _Michal Paulovic_, Sep 24 2023
%F A000796 From _Peter Bala_, Oct 28 2023: (Start)
%F A000796 Pi = 48*Sum_{n >= 0} (-1)^n/((6*n + 1)*(6*n + 3)*(6*n + 5)).
%F A000796 More generally, it appears that for k >= 0 we have Pi = A(k) + B(k)*Sum_{n >= 0} (-1)^n/((6*n + 1)*(6*n + 3)*...*(6*n + 6*k + 5)), where A(k) is a rational approximation to Pi and B(k) = (3 * 2^(3*k+3) * (3*k + 2)!) / (2^(3*k+1) - (-1)^k). The first few values of A(k) for k >= 0 are [0, 256/85, 65536/20955, 821559296/261636375, 6308233216/2008080987, 908209489444864/289093830828075, ...].
%F A000796 Pi = 16/5 - (288/5)*Sum_{n >= 0} (-1)^n * (6*n + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 9)).
%F A000796 More generally, it appears that for k >= 0 we have Pi = C(k) + D(k)*Sum_{n >= 0} (-1)^n* (6*n + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 6*k + 3)), where C(k) and D(k) are rational numbers. The case k = 0 is the Madhava-Gregory-Leibniz series for Pi.
%F A000796 Pi = 168/53 + (288/53)*Sum_{n >= 0} (-1)^n * (42*n^2 + 25*n)/((6*n + 1)*(6*n + 3)*(6*n + 5)*(6*n + 7)).
%F A000796 More generally, it appears that for k >= 1 we have Pi = E(k) + F(k)*Sum_{n >= 0} (-1)^n * (6*(6*k + 1)*n^2 + (24*k + 1)*n)/((6*n + 1)*(6*n + 3)*...*(6*n + 6*k + 1)), where E(k) and F(k) are rational numbers. (End)
%F A000796 From _Peter Bala_, Nov 10 2023: (Start)
%F A000796 The series representation Pi = 4 * Sum_{k >= 0} 1/(4*k + 1) - 1/(4*k + 3) given above by _Alexander R. Povolotsky_, Dec 25 2008, is the case n = 0 of the more general result (obtained by the WZ method): for n >= 0, there holds
%F A000796 Pi = Sum_{j = 0.. n-1} 2^(j+1)/((2*j + 1)*binomial(2*j,j)) + 8*(n+1)!*Sum_{k >= 0} 1/((4*k + 1)*(4*k + 3)*...*(4*k + 2*n + 3)).
%F A000796 Letting n -> oo gives the rapidly converging series Pi = Sum_{j >= 0} 2^(j+1)/((2*j + 1)*binomial(2*j,j)) due to Euler.
%F A000796 More generally, it appears that for n >= 1, Pi = 1/(2*n-1)!!^2 * Sum_{j >= 0} (Product_{i = 0..2*n-1} j - i) * 2^(j+1)/((2*j + 1)*binomial(2*j,j)).
%F A000796 For any integer n, Pi = (-1)^n * 4 * Sum_{k >= 0} 1/(4*k + 1 + 2*n) - 1/(4*k + 3 - 2*n). (End)
%F A000796 Pi = Product_{k>=1} ((k^3*(k + 2)*(2*k + 1)^2)/((k + 1)^4*(2*k - 1)^2))^k. - _Antonio Graciá Llorente_, Jun 13 2024
%F A000796 Equals Integral_{x=0..2} sqrt(8 - x^2) dx - 2 (see Ambrisi and Rizzi). - _Stefano Spezia_, Jul 21 2024
%F A000796 Equals 3 + 4*Sum_{k>0} (-1)^(k+1)/(4*k*(1+k)*(1+2*k)) (see Wells at p. 53). - _Stefano Spezia_, Aug 31 2024
%F A000796 Equals 4*Integral_{x=0..1} sqrt(1 - x^2) dx = lim_{n->oo} (4/n^2)*Sum_{k=0..n} sqrt(n^2 - k^2) (see Finch). - _Stefano Spezia_, Oct 19 2024
%F A000796 Equals Beta(1/2,1/2) (see Shamos). - _Stefano Spezia_, Jun 03 2025
%F A000796 From _Kritsada Moomuang_, Jun 18 2025: (Start)
%F A000796 Equals 2 + Integral_{x=0..1} 1/(sqrt(x)*(1 + sqrt(1 - x))) dx.
%F A000796 Equals 2 + Integral_{x=0..1} log(1 + sqrt(1 - x))/sqrt(x) dx. (End)
%F A000796 Pi = 2*arccos(1/phi) + arccos(1/phi^3) = 4*arcsin(1/phi) + 2*arcsin(1/phi^3) where phi = (1+sqrt(5))/2. - _Chittaranjan Pardeshi_, Jul 02 2025
%F A000796 Pi = Sum_{n >= 0} zeta(2*n)*(2^(2*n - 1) - 1)/2^(4*n - 3). - _Andrea Pinos_, Jul 29 2025
%e A000796 3.1415926535897932384626433832795028841971693993751058209749445923078164062\
%e A000796 862089986280348253421170679821480865132823066470938446095505822317253594081\
%e A000796 284811174502841027019385211055596446229489549303819...
%p A000796 Digits := 110: Pi*10^104:
%p A000796 ListTools:-Reverse(convert(floor(%), base, 10)); # _Peter Luschny_, Oct 29 2019
%t A000796 RealDigits[ N[ Pi, 105]] [[1]]
%t A000796 Table[ResourceFunction["NthDigit"][Pi, n], {n, 1, 102}] (* _Joan Ludevid_, Jun 22 2022; easy to compute a(10000000)=7 with this function; requires Mathematica 12.0+ *)
%o A000796 (Macsyma) py(x) := if equal(6,6+x^2) then 2*x else (py(x:x/3),3*%%-4*(%%-x)^3); py(3.); py(dfloat(%)); block([bfprecision:35], py(bfloat(%))) /* _Bill Gosper_, Sep 09 2002 */
%o A000796 (PARI) { default(realprecision, 20080); x=Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b000796.txt", n, " ", d)); } \\ _Harry J. Smith_, Apr 15 2009
%o A000796 (PARI) A796=[]; A000796(n)={if(n>#A796, localprec(n*6\5+29); A796=digits(Pi\.1^(precision(Pi)-3))); A796[n]} \\ NOTE: as the other programs, this returns the n-th term of the sequence, with n = 1, 2, 3, ... and not n = 1, 0, -1, -2, .... - _M. F. Hasler_, Jun 21 2022
%o A000796 (PARI) first(n)= default(realprecision, n+10); digits(floor(Pi*10^(n-1))) \\ _David A. Corneth_, Jun 21 2022
%o A000796 (PARI) lista(nn, p=20)= {my(u=10^(nn+p+1), f(x, u)=my(n=1, q=u\x, r=q, s=1, t); while(t=(q\=(x*x))\(n+=2), r+=(s=-s)*t); r*4); digits((4*f(5, u)-f(239, u))\10^(p+2)); } \\ Machin-like, with p > the maximal number of consecutive 9-digits to be expected (A048940) - _Ruud H.G. van Tol_, Dec 26 2024
%o A000796 (Haskell) -- see link: Literate Programs
%o A000796 import Data.Char (digitToInt)
%o A000796 a000796 n = a000796_list (n + 1) !! (n + 1)
%o A000796 a000796_list len = map digitToInt $ show $ machin' `div` (10 ^ 10) where
%o A000796 machin' = 4 * (4 * arccot 5 unity - arccot 239 unity)
%o A000796 unity = 10 ^ (len + 10)
%o A000796 arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where
%o A000796 arccot' x unity summa xpow n sign
%o A000796 | term == 0 = summa
%o A000796 | otherwise = arccot'
%o A000796 x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign)
%o A000796 where term = xpow `div` n
%o A000796 -- _Reinhard Zumkeller_, Nov 24 2012
%o A000796 (Haskell) -- See Niemeijer link and also Gibbons link.
%o A000796 a000796 n = a000796_list !! (n-1) :: Int
%o A000796 a000796_list = map fromInteger $ piStream (1, 0, 1)
%o A000796 [(n, a*d, d) | (n, d, a) <- map (\k -> (k, 2 * k + 1, 2)) [1..]] where
%o A000796 piStream z xs'@(x:xs)
%o A000796 | lb /= approx z 4 = piStream (mult z x) xs
%o A000796 | otherwise = lb : piStream (mult (10, -10 * lb, 1) z) xs'
%o A000796 where lb = approx z 3
%o A000796 approx (a, b, c) n = div (a * n + b) c
%o A000796 mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f)
%o A000796 -- _Reinhard Zumkeller_, Jul 14 2013, Jun 12 2013
%o A000796 (Magma) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // _Bruno Berselli_, Mar 12 2013
%o A000796 (Python) from sympy import pi, N; print(N(pi, 1000)) # _David Radcliffe_, Apr 10 2019
%o A000796 (Python)
%o A000796 from mpmath import mp
%o A000796 def A000796(n):
%o A000796 if n >= len(A000796.str): mp.dps = n*6//5+50; A000796.str = str(mp.pi-5/mp.mpf(10)**mp.dps)
%o A000796 return int(A000796.str[n if n>1 else 0])
%o A000796 A000796.str = '' # _M. F. Hasler_, Jun 21 2022
%o A000796 (SageMath)
%o A000796 m=125
%o A000796 x=numerical_approx(pi, digits=m+5)
%o A000796 a=[ZZ(i) for i in x.str(skip_zeroes=True) if i.isdigit()]
%o A000796 a[:m] # _G. C. Greubel_, Jul 18 2023
%Y A000796 Cf. A001203 (continued fraction).
%Y A000796 Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), this sequence (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A224750 (b=26), A224751 (b=27), A060707 (b=60). - _Jason Kimberley_, Dec 06 2012
%Y A000796 Decimal expansions of expressions involving Pi: A002388 (Pi^2), A003881 (Pi/4), A013661 (Pi^2/6), A019692 (2*Pi=tau), A019727 (sqrt(2*Pi)), A059956 (6/Pi^2), A060294 (2/Pi), A091925 (Pi^3), A092425 (Pi^4), A092731 (Pi^5), A092732 (Pi^6), A092735 (Pi^7), A092736 (Pi^8), A163973 (Pi/log(2)).
%Y A000796 Cf. A001901 (Pi/2; Wallis), A002736 (Pi^2/18; Euler), A007514 (Pi), A048581 (Pi; BBP), A054387 (Pi; Newton), A092798 (Pi/2), A096954 (Pi/4; Machin), A097486 (Pi), A122214 (Pi/2), A133766 (Pi/4 - 1/2), A133767 (5/6 - Pi/4), A166107 (Pi; MGL).
%Y A000796 Cf. A001622, A048940, A248682.
%K A000796 cons,nonn,nice,core,easy
%O A000796 1,1
%A A000796 _N. J. A. Sloane_
%E A000796 Additional comments from _William Rex Marshall_, Apr 20 2001