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A000796 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A000796 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A000796" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA000796">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A000796 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A000796 </div> <div class=seqname> Decimal expansion of Pi (or digits of Pi). <br><font size=-1>(Formerly M2218 N0880)</font> </div> </div> <div class=scorerefs> 1094 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>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, 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, 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</div> <div class=seqdatalinks> (<a href="/A000796/list">list</a>; <a href="/A000796/constant">constant</a>; <a href="/A000796/graph">graph</a>; <a href="/search?q=A000796+-id:A000796">refs</a>; <a href="/A000796/listen">listen</a>; <a href="/history?seq=A000796">history</a>; <a href="/search?q=id:A000796&fmt=text">text</a>; <a href="/A000796/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Sometimes called Archimedes's constant.</div> <div class=sectline>Ratio of a circle's circumference to its diameter.</div> <div class=sectline>Also area of a circle with radius 1.</div> <div class=sectline>Also surface area of a sphere with diameter 1.</div> <div class=sectline>A useful mnemonic for remembering the first few terms: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics ...</div> <div class=sectline>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. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Aug 09 2012</div> <div class=sectline>Also surface area of a quarter of a sphere of radius 1. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Oct 03 2013</div> <div class=sectline>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. - <a href="/wiki/User:Stanislav_Sykora">Stanislav Sykora</a>, Oct 31 2013</div> <div class=sectline>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 <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Aug 27 2014]</div> <div class=sectline>x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Dec 25 2013</div> <div class=sectline>Also diameter of a sphere whose surface area equals the volume of the circumscribed cube. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Jan 13 2014</div> <div class=sectline>From <a href="/wiki/User:Daniel_Forgues">Daniel Forgues</a>, Mar 20 2015: (Start)</div> <div class=sectline>An interesting anecdote about the base-10 representation of Pi, with 3 (integer part) as first (index 1) digit:</div> <div class=sectline> 358 0</div> <div class=sectline> 359 3</div> <div class=sectline> 360 6</div> <div class=sectline> 361 0</div> <div class=sectline> 362 0</div> <div class=sectline>And the circle is customarily subdivided into 360 degrees (although Pi radians yields half the circle)...</div> <div class=sectline>(End)</div> <div class=sectline>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. - <a href="/wiki/User:Martin_Renner">Martin Renner</a>, Sep 07 2016</div> <div class=sectline>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". - <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Jan 23 2019</div> <div class=sectline>On March 14, 2019, Emma Haruka Iwao announced the calculation of 31.4 trillion digits of Pi using Google Cloud's infrastructure. - <a href="/wiki/User:David_Radcliffe">David Radcliffe</a>, Apr 10 2019</div> <div class=sectline>Also volume of three quarters of a sphere of radius 1. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Aug 16 2019</div> <div class=sectline>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. - <a href="/wiki/User:Alonso_del_Arte">Alonso del Arte</a>, Aug 23 2021</div> <div class=sectline>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. - <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Jul 21 2023</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>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.</div> <div class=sectline>J. Arndt & C. Haenel, Pi Unleashed, Springer NY 2001.</div> <div class=sectline>P. Beckmann, A History of Pi, Golem Press, Boulder, CO, 1977.</div> <div class=sectline>J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997.</div> <div class=sectline>P. Eyard and J.-P. Lafon, The Number Pi, Amer. Math. Soc., 2004.</div> <div class=sectline>S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.4.</div> <div class=sectline>Le Petit Archimede, Special Issue On Pi, Supplement to No. 64-5, May 1980 ADCS Amiens.</div> <div class=sectline>Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 31.</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>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.</div> <div class=sectline>David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See pp. 48-55.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Harry J. Smith, <a href="/A000796/b000796.txt">Table of n, a(n) for n = 1..20000</a></div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>Dave Andersen, <a href="http://www.angio.net/pi/piquery">Pi-Search Page</a></div> <div class=sectline>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></div> <div class=sectline>Anonymous, <a href="http://mapage.noos.fr/echolalie/l127.htm">Liste de quelques milliers de decimales du nombre de pi</a></div> <div class=sectline>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]</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>Steve Baker and Thomas Moore, <a href="https://storage.googleapis.com/pi100t/index.html">100 trillion digits of pi</a></div> <div class=sectline>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.</div> <div class=sectline>J. M. Borwein, <a href="http://www.cecm.sfu.ca/~jborwein/pi_cover.html">Talking about Pi</a></div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>Christian Boyer, <a href="http://www.multimagie.com/">MultiMagic Squares</a></div> <div class=sectline>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]</div> <div class=sectline>D. Castellanos, <a href="http://www.jstor.org/stable/2690037">The ubiquitous pi</a>, Math. Mag., 61 (1988), 67-98 and 148-163.</div> <div class=sectline>Jonas Castillo Toloza, <a href="http://www.lifesmith.com/mathfun.html#41">Fascinating Method for Finding Pi</a></div> <div class=sectline>E. S. Croot, <a href="http://people.math.gatech.edu/~ecroot/transcend.pdf">Pade Approximations and the Transcendence of pi</a></div> <div class=sectline>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.</div> <div class=sectline>L. Euler, <a href="http://eulerarchive.maa.org/pages/E041.html">De summis serierum reciprocarum</a>, E41.</div> <div class=sectline>Eureka, <a href="http://users.skynet.be/ekurea/toutpi.html">Tout pi or not tout pi</a></div> <div class=sectline>Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/publist.html">Zeta function expansions of some classical constants</a></div> <div class=sectline>Jeremy Gibbons, <a href="http://www.cs.ox.ac.uk/jeremy.gibbons/publications/spigot.pdf">Unbounded Spigot Algorithms for the Digits of Pi</a></div> <div class=sectline>GJ, <a href="http://web.archive.org/web/20011214030954/http://gj.mit.edu/pi/digits/10million.txt">10 million digits of Pi</a></div> <div class=sectline>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]</div> <div class=sectline>Xavier Gourdon, <a href="http://numbers.computation.free.fr/Constants/Algorithms/nthdigit.html">A new algorithm for computing Pi in base 10</a></div> <div class=sectline>X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Pi/pi.html">Archimedes' constant Pi</a></div> <div class=sectline>B. Gourevitch, <a href="http://www.pi314.net">L'univers de Pi</a></div> <div class=sectline>Antonio Gracia Llorente, <a href="https://osf.io/preprints/osf/dg8tf">Novel Infinite Products πe and π/e</a>, OSF Preprint, 2024.</div> <div class=sectline>L. Grebelius, <a href="http://web.archive.org/web/20130303114650/http://www2.tripnet.se/~nlg/pi0001.htm">Approximation of Pi: First 1000000 digits</a></div> <div class=sectline>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).</div> <div class=sectline>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.</div> <div class=sectline>H. Havermann, <a href="https://web.archive.org/web/20181130124011/http://chesswanks.com/pxp/cfpi.html">Simple Continued Fraction for Pi</a> [archived page]</div> <div class=sectline>M. D. Huberty et al., <a href="http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html">100000 Digits of Pi</a></div> <div class=sectline>ICON Project, <a href="https://www2.cs.arizona.edu/icon/oddsends/pi.htm">Pi to 50000 places</a> [archived page]</div> <div class=sectline>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></div> <div class=sectline>P. Johns, <a href="https://web.archive.org/web/20020207043745/http://www.wpdpi.com:80/pi.shtml">120000 Digits of Pi</a> [archived page]</div> <div class=sectline>Yasumasa Kanada, <a href="http://www.super-computing.org/">1.24 trillion digits of Pi</a></div> <div class=sectline>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]</div> <div class=sectline>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]</div> <div class=sectline>Johannes W. Meijer, <a href="/A000796/a000796.jpg">Pi everywhere</a> poster, Mar 14 2013</div> <div class=sectline>J. Moyer, <a href="http://www.rsok.com/~jrm/pi10000.txt">First 10000 digits of pi</a></div> <div class=sectline>NERSC, <a href="http://pi.nersc.gov/">Search Pi</a> [broken link]</div> <div class=sectline>Remco Niemeijer, <a href="http://programmingpraxis.com/2009/02/20/the-digits-of-pi/">The Digits of Pi</a>, programmingpraxis.</div> <div class=sectline>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]</div> <div class=sectline>Chittaranjan Pardeshi, <a href="/A000796/a000796.pdf">BBP-Like formula for Pi in Golden Ratio Base Phi</a></div> <div class=sectline>Michael Penn, <a href="https://www.youtube.com/watch?v=dzzbhfudx5M">A nice inverse tangent integral.</a>, YouTube video, 2020.</div> <div class=sectline>Michael Penn, <a href="https://www.youtube.com/watch?v=dFKbVTHK4tU">Pi is irrational (π∉ℚ)</a>, YouTube video, 2020.</div> <div class=sectline>I. Peterson, <a href="http://web.cs.ucla.edu/~klinger/mathland_3_11.html">A Passion for Pi</a></div> <div class=sectline>G. M. Phillips, <a href="http://www.mcs.st-and.ac.uk/~gmp/gmpCON.html">Table of contents of "Pi: A source Book"</a></div> <div class=sectline>Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/pi10000.txt">10000 digits of Pi</a></div> <div class=sectline>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.</div> <div class=sectline>D. Pothet, <a href="http://perso.wanadoo.fr/didier.pothet/pi.html">Chronologie du calcul des decimales de pi</a> [broken link]</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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></div> <div class=sectline>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.</div> <div class=sectline>Grant Sanderson, <a href="https://www.youtube.com/watch?v=jsYwFizhncE">Why do colliding blocks compute pi?</a>, 3Blue1Brown video (2019).</div> <div class=sectline>Daniel B. Sedory, <a href="http://thestarman.pcministry.com/math/pi/index.html">The Pi Pages</a></div> <div class=sectline>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.</div> <div class=sectline>Jean-Louis Sigrist, <a href="http://jlsigrist.com/pi.html">Les 128000 premieres decimales du nombre PI</a></div> <div class=sectline>Sizes, <a href="http://www.sizes.com/numbers/pi.htm">pi</a></div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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.</div> <div class=sectline>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]</div> <div class=sectline>Wislawa Szymborska, <a href="http://katherinestange.com/mathweb/p_p2.html">Pi (The admirable number Pi)</a>, Miracle Fair, 2002.</div> <div class=sectline>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.</div> <div class=sectline>Stan Wagon, <a href="http://pi314.at/math/normal.html">Is Pi Normal?</a></div> <div class=sectline>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></div> <div class=sectline>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></div> <div class=sectline>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></div> <div class=sectline>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></div> <div class=sectline><a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a></div> <div class=sectline><a href="/index/Cor#core">Index entries for "core" sequences</a></div> <div class=sectline><a href="/index/Tra#transcendental">Index entries for transcendental numbers</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>Pi = 4*Sum_{k>=0} (-1)^k/(2k+1) [Madhava-Gregory-Leibniz, 1450-1671]. - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Feb 27 2013</div> <div class=sectline>From <a href="/wiki/User:Johannes_W._Meijer">Johannes W. Meijer</a>, Mar 10 2013: (Start)</div> <div class=sectline>2/Pi = (sqrt(2)/2) * (sqrt(2 + sqrt(2))/2) * (sqrt(2 + sqrt(2 + sqrt(2)))/2) * ... [Viete, 1593]</div> <div class=sectline>2/Pi = Product_{k>=1} (4*k^2-1)/(4*k^2). [Wallis, 1655]</div> <div class=sectline>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]</div> <div class=sectline>Pi/4 = 4*arctan(1/5) - arctan(1/239). [Machin, 1706]</div> <div class=sectline>Pi^2/6 = 3*Sum_{n>=1} 1/(n^2*binomial(2*n,n)). [Euler, 1748]</div> <div class=sectline>1/Pi = (2*sqrt(2)/9801) * Sum_{n>=0} (4*n)!*(1103+26390*n)/((n!)^4*396^(4*n)). [Ramanujan, 1914]</div> <div class=sectline>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]</div> <div class=sectline>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)</div> <div class=sectline>Pi = 4 * Sum_{k>=0} 1/(4*k+1) - 1/(4*k+3). - <a href="/wiki/User:Alexander_R._Povolotsky">Alexander R. Povolotsky</a>, Dec 25 2008</div> <div class=sectline>Pi = 4*sqrt(-1*(Sum_{n>=0} (i^(2*n+1))/(2*n+1))^2). - <a href="/wiki/User:Alexander_R._Povolotsky">Alexander R. Povolotsky</a>, Jan 25 2009</div> <div class=sectline>Pi = Integral_{x=-infinity..infinity} dx/(1+x^2). - <a href="/wiki/User:Mats_Granvik">Mats Granvik</a> and <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Sep 23 2012</div> <div class=sectline>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)). - <a href="/wiki/User:José_de_Jesús_Camacho_Medina">José de Jesús Camacho Medina</a>, Jan 20 2014</div> <div class=sectline>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. - <a href="/wiki/User:Dimitris_Valianatos">Dimitris Valianatos</a>, May 05 2016</div> <div class=sectline>From <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, Aug 07 2016: (Start)</div> <div class=sectline>Pi = Sum_{k>=1} (3^k - 1)*zeta(k+1)/4^k.</div> <div class=sectline>Pi = 2*Product_{k>=2} sec(Pi/2^k).</div> <div class=sectline>Pi = 2*Integral_{x>=0} sin(x)/x dx. (End)</div> <div class=sectline>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). - <a href="/wiki/User:Sanjar_Abrarov">Sanjar Abrarov</a>, Feb 07 2017</div> <div class=sectline>Pi = Integral_{x = 0..2} sqrt(x/(2 - x)) dx. - <a href="/wiki/User:Arkadiusz_Wesolowski">Arkadiusz Wesolowski</a>, Nov 20 2017</div> <div class=sectline>Pi = lim_{n->infinity} 2/n * Sum_{m=1,n} ( sqrt( (n+1)^2 - m^2 ) - sqrt( n^2 - m^2 ) ). - <a href="/wiki/User:Dimitri_Papadopoulos">Dimitri Papadopoulos</a>, May 31 2019</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Oct 29 2019: (Start)</div> <div class=sectline>Pi = Sum_{n >= 0} 2^(n+1)/( binomial(2*n,n)*(2*n + 1) ) - Euler.</div> <div class=sectline>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.</div> <div class=sectline>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 <a href="/A085565" title="Decimal expansion of lemniscate constant A.">A085565</a> (resp. <a href="/A076390" title="Decimal expansion of lemniscate constant B.">A076390</a>). (End)</div> <div class=sectline>Pi = Im(log(-i^i)) = log(i^i)*(-2). - <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Oct 29 2019</div> <div class=sectline>From <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Aug 15 2020: (Start)</div> <div class=sectline>Equals 2 + Integral_{x=0..1} arccos(x)^2 dx.</div> <div class=sectline>Equals Integral_{x=0..oo} log(1 + 1/x^2) dx.</div> <div class=sectline>Equals Integral_{x=0..oo} log(1 + x^2)/x^2 dx.</div> <div class=sectline>Equals Integral_{x=-oo..oo} exp(x/2)/(exp(x) + 1) dx. (End)</div> <div class=sectline>Equals 4*(1/2)!^2 = 4*Gamma(3/2)^2. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Aug 23 2021</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Dec 08 2021: (Start)</div> <div class=sectline>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)).</div> <div class=sectline>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)).</div> <div class=sectline>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).</div> <div class=sectline>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).</div> <div class=sectline>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)</div> <div class=sectline>For odd n, Pi = (2^(n-1)/<a href="/A001818" title="Squares of double factorials: (1*3*5*...*(2n-1))^2 = ((2*n-1)!!)^2.">A001818</a>((n-1)/2))*gamma(n/2)^2. - <a href="/wiki/User:Alan_Michael_Gómez_Calderón">Alan Michael Gómez Calderón</a>, Mar 11 2022</div> <div class=sectline>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. - <a href="/wiki/User:Chittaranjan_Pardeshi">Chittaranjan Pardeshi</a>, May 16 2022</div> <div class=sectline>Pi = sqrt(3)*(27*S - 36)/24, where S = <a href="/A248682" title="Decimal expansion of Sum_{n >= 0} (floor(n/2)!)^2/n!.">A248682</a>. - <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Jul 22 2022</div> <div class=sectline>Equals Integral_{x=0..1} 1/sqrt(x-x^2) dx. - <a href="/wiki/User:Michal_Paulovic">Michal Paulovic</a>, Sep 24 2023</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Oct 28 2023: (Start)</div> <div class=sectline>Pi = 48*Sum_{n >= 0} (-1)^n/((6*n + 1)*(6*n + 3)*(6*n + 5)).</div> <div class=sectline>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, ...].</div> <div class=sectline>Pi = 16/5 - (288/5)*Sum_{n >= 0} (-1)^n * (6*n + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 9)).</div> <div class=sectline>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.</div> <div class=sectline>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)).</div> <div class=sectline>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)</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Nov 10 2023: (Start)</div> <div class=sectline>The series representation Pi = 4 * Sum_{k >= 0} 1/(4*k + 1) - 1/(4*k + 3) given above by <a href="/wiki/User:Alexander_R._Povolotsky">Alexander R. Povolotsky</a>, Dec 25 2008, is the case n = 0 of the more general result (obtained by the WZ method): for n >= 0, there holds</div> <div class=sectline>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)).</div> <div class=sectline>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.</div> <div class=sectline>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)).</div> <div class=sectline>For any integer n, Pi = (-1)^n * 4 * Sum_{k >= 0} 1/(4*k + 1 + 2*n) - 1/(4*k + 3 - 2*n). (End)</div> <div class=sectline>Pi = Product_{k>=1} ((k^3*(k + 2)*(2*k + 1)^2)/((k + 1)^4*(2*k - 1)^2))^k. - <a href="/wiki/User:Antonio_Graciá_Llorente">Antonio Graciá Llorente</a>, Jun 13 2024</div> <div class=sectline>Equals Integral_{x=0..2} sqrt(8 - x^2) dx - 2 (see Ambrisi and Rizzi). - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Jul 21 2024</div> <div class=sectline>Equals 3 + 4*Sum_{k>0} (-1)^(k+1)/(4*k*(1+k)*(1+2*k)) (see Wells at p. 53). - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Aug 31 2024</div> <div class=sectline>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). - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Oct 19 2024</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>3.1415926535897932384626433832795028841971693993751058209749445923078164062\</div> <div class=sectline>862089986280348253421170679821480865132823066470938446095505822317253594081\</div> <div class=sectline>284811174502841027019385211055596446229489549303819...</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>Digits := 110: Pi*10^104:</div> <div class=sectline>ListTools:-Reverse(convert(floor(%), base, 10)); # <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Oct 29 2019</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>RealDigits[ N[ Pi, 105]] [[1]]</div> <div class=sectline>Table[ResourceFunction["NthDigit"][Pi, n], {n, 1, 102}] (* <a href="/wiki/User:Joan_Ludevid">Joan Ludevid</a>, Jun 22 2022; easy to compute a(10000000)=7 with this function; requires Mathematica 12.0+ *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(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(%))) /* <a href="/wiki/User:Bill_Gosper">Bill Gosper</a>, Sep 09 2002 */</div> <div class=sectline>(PARI) { default(realprecision, 20080); x=Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b000796.txt", n, " ", d)); } \\ <a href="/wiki/User:Harry_J._Smith">Harry J. Smith</a>, Apr 15 2009</div> <div class=sectline>(PARI) A796=[]; <a href="/A000796" title="Decimal expansion of Pi (or digits of Pi).">A000796</a>(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, .... - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Jun 21 2022</div> <div class=sectline>(PARI) first(n)= default(realprecision, n+10); digits(floor(Pi*10^(n-1))) \\ <a href="/wiki/User:David_A._Corneth">David A. Corneth</a>, Jun 21 2022</div> <div class=sectline>(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 (<a href="/A048940" title="Starting position of the first occurrence of a string of at least n '9's in the decimal expansion of Pi.">A048940</a>) - <a href="/wiki/User:Ruud_H.G._van_Tol">Ruud H.G. van Tol</a>, Dec 26 2024</div> <div class=sectline>(Haskell) -- see link: Literate Programs</div> <div class=sectline>import Data.Char (digitToInt)</div> <div class=sectline>a000796 n = a000796_list (n + 1) !! (n + 1)</div> <div class=sectline>a000796_list len = map digitToInt $ show $ machin' `div` (10 ^ 10) where</div> <div class=sectline> machin' = 4 * (4 * arccot 5 unity - arccot 239 unity)</div> <div class=sectline> unity = 10 ^ (len + 10)</div> <div class=sectline> arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where</div> <div class=sectline> arccot' x unity summa xpow n sign</div> <div class=sectline> | term == 0 = summa</div> <div class=sectline> | otherwise = arccot'</div> <div class=sectline> x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign)</div> <div class=sectline> where term = xpow `div` n</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Nov 24 2012</div> <div class=sectline>(Haskell) -- See Niemeijer link and also Gibbons link.</div> <div class=sectline>a000796 n = a000796_list !! (n-1) :: Int</div> <div class=sectline>a000796_list = map fromInteger $ piStream (1, 0, 1)</div> <div class=sectline> [(n, a*d, d) | (n, d, a) <- map (\k -> (k, 2 * k + 1, 2)) [1..]] where</div> <div class=sectline> piStream z xs'@(x:xs)</div> <div class=sectline> | lb /= approx z 4 = piStream (mult z x) xs</div> <div class=sectline> | otherwise = lb : piStream (mult (10, -10 * lb, 1) z) xs'</div> <div class=sectline> where lb = approx z 3</div> <div class=sectline> approx (a, b, c) n = div (a * n + b) c</div> <div class=sectline> mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f)</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 14 2013, Jun 12 2013</div> <div class=sectline>(Magma) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Mar 12 2013</div> <div class=sectline>(Python) from sympy import pi, N; print(N(pi, 1000)) # <a href="/wiki/User:David_Radcliffe">David Radcliffe</a>, Apr 10 2019</div> <div class=sectline>(Python)</div> <div class=sectline>from mpmath import mp</div> <div class=sectline>def <a href="/A000796" title="Decimal expansion of Pi (or digits of Pi).">A000796</a>(n):</div> <div class=sectline> if n >= len(<a href="/A000796" title="Decimal expansion of Pi (or digits of Pi).">A000796</a>.str): mp.dps = n*6//5+50; <a href="/A000796" title="Decimal expansion of Pi (or digits of Pi).">A000796</a>.str = str(mp.pi-5/mp.mpf(10)**mp.dps)</div> <div class=sectline> return int(<a href="/A000796" title="Decimal expansion of Pi (or digits of Pi).">A000796</a>.str[n if n>1 else 0])</div> <div class=sectline><a href="/A000796" title="Decimal expansion of Pi (or digits of Pi).">A000796</a>.str = '' # <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Jun 21 2022</div> <div class=sectline>(SageMath)</div> <div class=sectline>m=125</div> <div class=sectline>x=numerical_approx(pi, digits=m+5)</div> <div class=sectline>a=[ZZ(i) for i in x.str(skip_zeroes=True) if i.isdigit()]</div> <div class=sectline>a[:m] # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Jul 18 2023</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A001203" title="Simple continued fraction expansion of Pi.">A001203</a> (continued fraction).</div> <div class=sectline>Pi in base b: <a href="/A004601" title="Expansion of Pi in base 2 (or, binary expansion of Pi).">A004601</a> (b=2), <a href="/A004602" title="Expansion of Pi in base 3.">A004602</a> (b=3), <a href="/A004603" title="Expansion of Pi in base 4.">A004603</a> (b=4), <a href="/A004604" title="Expansion of Pi in base 5.">A004604</a> (b=5), <a href="/A004605" title="Expansion of Pi in base 6.">A004605</a> (b=6), <a href="/A004606" title="Expansion of Pi in base 7.">A004606</a> (b=7), <a href="/A006941" title="Expansion of Pi in base 8.">A006941</a> (b=8), <a href="/A004608" title="Expansion of Pi in base 9.">A004608</a> (b=9), this sequence (b=10), <a href="/A068436" title="Expansion of Pi in base 11.">A068436</a> (b=11), <a href="/A068437" title="Expansion of Pi in base 12.">A068437</a> (b=12), <a href="/A068438" title="Expansion of Pi in base 13.">A068438</a> (b=13), <a href="/A068439" title="Expansion of Pi in base 14.">A068439</a> (b=14), <a href="/A068440" title="Expansion of Pi in base 15.">A068440</a> (b=15), <a href="/A062964" title="Pi in hexadecimal.">A062964</a> (b=16), <a href="/A224750" title="Expansion of Pi in base 26.">A224750</a> (b=26), <a href="/A224751" title="Expansion of Pi in base 27.">A224751</a> (b=27), <a href="/A060707" title="Base-60 (Babylonian or sexagesimal) expansion of Pi.">A060707</a> (b=60). - <a href="/wiki/User:Jason_Kimberley">Jason Kimberley</a>, Dec 06 2012</div> <div class=sectline>Decimal expansions of expressions involving Pi: <a href="/A002388" title="Decimal expansion of Pi^2.">A002388</a> (Pi^2), <a href="/A003881" title="Decimal expansion of Pi/4.">A003881</a> (Pi/4), <a href="/A013661" title="Decimal expansion of Pi^2/6 = zeta(2) = Sum_{m>=1} 1/m^2.">A013661</a> (Pi^2/6), <a href="/A019692" title="Decimal expansion of 2*Pi.">A019692</a> (2*Pi=tau), <a href="/A019727" title="Decimal expansion of sqrt(2*Pi).">A019727</a> (sqrt(2*Pi)), <a href="/A059956" title="Decimal expansion of 6/Pi^2.">A059956</a> (6/Pi^2), <a href="/A060294" title="Decimal expansion of Buffon's constant 2/Pi.">A060294</a> (2/Pi), <a href="/A091925" title="Decimal expansion of Pi^3.">A091925</a> (Pi^3), <a href="/A092425" title="Decimal expansion of Pi^4.">A092425</a> (Pi^4), <a href="/A092731" title="Decimal expansion of Pi^5.">A092731</a> (Pi^5), <a href="/A092732" title="Decimal expansion of Pi^6.">A092732</a> (Pi^6), <a href="/A092735" title="Decimal expansion of Pi^7.">A092735</a> (Pi^7), <a href="/A092736" title="Decimal expansion of Pi^8.">A092736</a> (Pi^8), <a href="/A163973" title="Decimal expansion of Van der Pauw's constant = Pi/log(2).">A163973</a> (Pi/log(2)).</div> <div class=sectline>Cf. <a href="/A001901" title="Successive numerators of Wallis's approximation to Pi/2 (reduced).">A001901</a> (Pi/2; Wallis), <a href="/A002736" title="Apéry numbers: a(n) = n^2*C(2n,n).">A002736</a> (Pi^2/18; Euler), <a href="/A007514" title="Pi = Sum_{n >= 0} a(n)/n!.">A007514</a> (Pi), <a href="/A048581" title="Numerators of b(n) = (1/16^n)*(4/(8*n+1) - 2/(8*n+4) - 1/(8*n+5) - 1/(8*n+6)).">A048581</a> (Pi; BBP), <a href="/A054387" title="Numerators of coefficients of 1/2^(2n+1) in Newton's series for Pi.">A054387</a> (Pi; Newton), <a href="/A092798" title="Numerator of partial products in an approximation of Pi/2.">A092798</a> (Pi/2), <a href="/A096954" title="Numerators of rational approximation to Pi/4 from Machins's formula.">A096954</a> (Pi/4; Machin), <a href="/A097486" title="A relationship between Pi and the Mandelbrot set. a(n) = number of iterations of z^2 + c that c-values -0.75 + x*i go throug...">A097486</a> (Pi), <a href="/A122214" title="Numerators in infinite products for Pi/2, e and e^gamma (reduced).">A122214</a> (Pi/2), <a href="/A133766" title="a(n) = (4*n+1)*(4*n+3)*(4*n+5).">A133766</a> (Pi/4 - 1/2), <a href="/A133767" title="a(n) = (4*n+3)*(4*n+5)*(4*n+7).">A133767</a> (5/6 - Pi/4), <a href="/A166107" title="A sequence related to the Madhava-Gregory-Leibniz formula for Pi.">A166107</a> (Pi; MGL).</div> <div class=sectline>Cf. <a href="/A048940" title="Starting position of the first occurrence of a string of at least n '9's in the decimal expansion of Pi.">A048940</a>, <a href="/A248682" title="Decimal expansion of Sum_{n >= 0} (floor(n/2)!)^2/n!.">A248682</a>.</div> <div class=sectline>Sequence in context: <a href="/A247385" title="Erroneous decimal expansion (or digits) of Pi engraved in the wall of the Robertson Tunnel at the Washington Park (MAX stati...">A247385</a> <a href="/A253214" title="Decimal expansion of log(640320^3)/sqrt(163), a Ramanujan constant producing 16 correct digits of Pi.">A253214</a> <a href="/A112602" title="Erroneous version of decimal expansion of Pi (see A000796 for the correct version).">A112602</a> * <a href="/A212131" title="Decimal expansion of k such that e^(k*sqrt(163)) = round(e^(Pi*sqrt(163))).">A212131</a> <a href="/A379323" title="Decimal expansion of log((2*5! + (8 - 1)!)^sqrt(9) + 4! + (3!)!)/sqrt(67).">A379323</a> <a href="/A379276" title="Decimal expansion of 2^(5^0.4) - 0.6 - ((0.3^9)/7)^(0.8^0.1).">A379276</a></div> <div class=sectline>Adjacent sequences: <a href="/A000793" title="Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.">A000793</a> <a href="/A000794" title="Permanent of projective plane of order n.">A000794</a> <a href="/A000795" title="Salié numbers: expansion of cosh x / cos x = Sum_{n >= 0} a(n)*x^(2n)/(2n)!.">A000795</a> * <a href="/A000797" title="Numbers that are not the sum of 4 tetrahedral numbers.">A000797</a> <a href="/A000798" title="Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.">A000798</a> <a href="/A000799" title="a(n) = floor(2^n / n).">A000799</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a decimal expansion of a number"><a href="/A000796/constant">cons</a></span>,<span title="a sequence of nonnegative numbers">nonn</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="an important sequence">core</span>,<span title="it is very easy to produce terms of sequence">easy</span>,<span title="edited within the last two weeks">changed</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Additional comments from <a href="/wiki/User:William_Rex_Marshall">William Rex Marshall</a>, Apr 20 2001</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified February 22 20:36 EST 2025. Contains 381121 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>