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%I M2218 N0880 #649 Apr 12 2024 20:17:57
%S 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 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 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 Decimal expansion of Pi (or digits of Pi).
%C Sometimes called Archimedes's constant.
%C Ratio of a circle's circumference to its diameter.
%C Also area of a circle with radius 1.
%C Also surface area of a sphere with diameter 1.
%C 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 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 Also surface area of a quarter of a sphere of radius 1. - _Omar E. Pol_, Oct 03 2013
%C 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 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 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 Also diameter of a sphere whose surface area equals the volume of the circumscribed cube. - _Omar E. Pol_, Jan 13 2014
%C From _Daniel Forgues_, Mar 20 2015: (Start)
%C An interesting anecdote about the base-10 representation of Pi, with 3 (integer part) as first (index 1) digit:
%C 358 0
%C 359 3
%C 360 6
%C 361 0
%C 362 0
%C And the circle is customarily subdivided into 360 degrees (although Pi radians yields half the circle)...
%C (End)
%C 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 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 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 Also volume of three quarters of a sphere of radius 1. - _Omar E. Pol_, Aug 16 2019
%C 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 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
%D 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 J. Arndt & C. Haenel, Pi Unleashed, Springer NY 2001.
%D P. Beckmann, A History of Pi, Golem Press, Boulder, CO, 1977.
%D J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997.
%D P. Eyard and J.-P. Lafon, The Number Pi, Amer. Math. Soc., 2004.
%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.4.
%D Le Petit Archimede, Special Issue On Pi, Supplement to No. 64-5, May 1980 ADCS Amiens.
%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 31.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Harry J. Smith, <a href="/A000796/b000796.txt">Table of n, a(n) for n = 1..20000</a>
%H 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 Dave Andersen, <a href="http://www.angio.net/pi/piquery">Pi-Search Page</a>
%H 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 Anonymous, <a href="http://mapage.noos.fr/echolalie/l127.htm">Liste de quelques milliers de decimales du nombre de pi</a>
%H 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 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 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 Steve Baker and Thomas Moore, <a href="https://storage.googleapis.com/pi100t/index.html">100 trillion digits of pi</a>
%H 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 J. M. Borwein, <a href="http://www.cecm.sfu.ca/~jborwein/pi_cover.html">Talking about Pi</a>
%H 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 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 Christian Boyer, <a href="http://www.multimagie.com/">MultiMagic Squares</a>
%H 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 D. Castellanos, <a href="http://www.jstor.org/stable/2690037">The ubiquitous pi</a>, Math. Mag., 61 (1988), 67-98 and 148-163.
%H Jonas Castillo Toloza, <a href="http://www.lifesmith.com/mathfun.html#41">Fascinating Method for Finding Pi</a>
%H E. S. Croot, <a href="http://people.math.gatech.edu/~ecroot/transcend.pdf">Pade Approximations and the Transcendence of pi</a>
%H 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 L. Euler, <a href="http://eulerarchive.maa.org/pages/E041.html">De summis serierum reciprocarum</a>, E41.
%H Eureka, <a href="http://users.skynet.be/ekurea/toutpi.html">Tout pi or not tout pi</a>
%H Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/publist.html">Zeta function expansions of some classical constants</a>
%H 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 GJ, <a href="http://web.archive.org/web/20011214030954/http://gj.mit.edu/pi/digits/10million.txt">10 million digits of Pi</a>
%H 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 Xavier Gourdon, <a href="http://numbers.computation.free.fr/Constants/Algorithms/nthdigit.html">A new algorithm for computing Pi in base 10</a>
%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Pi/pi.html">Archimedes' constant Pi</a>
%H B. Gourevitch, <a href="http://www.pi314.net">L'univers de Pi</a>
%H 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 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 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 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 M. D. Huberty et al., <a href="http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html">100000 Digits of Pi</a>
%H ICON Project, <a href="https://www2.cs.arizona.edu/icon/oddsends/pi.htm">Pi to 50000 places</a> [archived page]
%H 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 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 Yasumasa Kanada, <a href="http://www.super-computing.org/">1.24 trillion digits of Pi</a>
%H 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 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 Johannes W. Meijer, <a href="/A000796/a000796.jpg">Pi everywhere</a> poster, Mar 14 2013
%H J. Moyer, <a href="http://www.rsok.com/~jrm/pi10000.txt">First 10000 digits of pi</a>
%H NERSC, <a href="http://pi.nersc.gov/">Search Pi</a> [broken link]
%H Remco Niemeijer, <a href="http://programmingpraxis.com/2009/02/20/the-digits-of-pi/">The Digits of Pi</a>, programmingpraxis.
%H 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 Chittaranjan Pardeshi, <a href="/A000796/a000796.pdf">BBP-Like formula for Pi in Golden Ratio Base Phi</a>
%H Michael Penn, <a href="https://www.youtube.com/watch?v=dzzbhfudx5M">A nice inverse tangent integral.</a>, YouTube video, 2020.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=dFKbVTHK4tU">Pi is irrational (π∉ℚ)</a>, YouTube video, 2020.
%H I. Peterson, <a href="http://web.cs.ucla.edu/~klinger/mathland_3_11.html">A Passion for Pi</a>
%H 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 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/pi10000.txt">10000 digits of Pi</a>
%H 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 D. Pothet, <a href="http://perso.wanadoo.fr/didier.pothet/pi.html">Chronologie du calcul des decimales de pi</a> [broken link]
%H 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 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 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 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 Grant Sanderson, <a href="https://www.youtube.com/watch?v=jsYwFizhncE">Why do colliding blocks compute pi?</a>, 3Blue1Brown video (2019).
%H Daniel B. Sedory, <a href="http://thestarman.pcministry.com/math/pi/index.html">The Pi Pages</a>
%H 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 Jean-Louis Sigrist, <a href="http://jlsigrist.com/pi.html">Les 128000 premieres decimales du nombre PI</a>
%H Sizes, <a href="http://www.sizes.com/numbers/pi.htm">pi</a>
%H 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 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 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 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 Wislawa Szymborska, <a href="http://katherinestange.com/mathweb/p_p2.html">Pi (The admirable number Pi)</a>, Miracle Fair, 2002.
%H 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 Stan Wagon, <a href="http://pi314.at/math/normal.html">Is Pi Normal?</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pi.html">Pi</a> and <a href="http://mathworld.wolfram.com/PiDigits.html">Pi Digits</a>
%H 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>, and <a href="https://www.wikipedia.org/wiki/Pi">Pi</a>
%H 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 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 <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Pi = 4*Sum_{k>=0} (-1)^k/(2k+1) [Madhava-Gregory-Leibniz, 1450-1671]. - _N. J. A. Sloane_, Feb 27 2013
%F From _Johannes W. Meijer_, Mar 10 2013: (Start)
%F 2/Pi = (sqrt(2)/2) * (sqrt(2 + sqrt(2))/2) * (sqrt(2 + sqrt(2 + sqrt(2)))/2) * ... [Viete, 1593]
%F 2/Pi = Product_{k>=1} (4*k^2-1)/(4*k^2). [Wallis, 1655]
%F 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 Pi/4 = 4*arctan(1/5) - arctan(1/239). [Machin, 1706]
%F Pi^2/6 = 3*Sum_{n>=1} 1/(n^2*binomial(2*n,n)). [Euler, 1748]
%F 1/Pi = (2*sqrt(2)/9801) * Sum_{n>=0} (4*n)!*(1103+26390*n)/((n!)^4*396^(4*n)). [Ramanujan, 1914]
%F 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 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 Pi = 4 * Sum_{k>=0} 1/(4*k+1) - 1/(4*k+3). - _Alexander R. Povolotsky_, Dec 25 2008
%F Pi = 4*sqrt(-1*(Sum_{n>=0} (i^(2*n+1))/(2*n+1))^2). - _Alexander R. Povolotsky_, Jan 25 2009
%F Pi = Integral_{x=-infinity..infinity} dx/(1+x^2). - _Mats Granvik_ and _Gary W. Adamson_, Sep 23 2012
%F 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 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 From _Ilya Gutkovskiy_, Aug 07 2016: (Start)
%F Pi = Sum_{k>=1} (3^k - 1)*zeta(k+1)/4^k.
%F Pi = 2*Product_{k>=2} sec(Pi/2^k).
%F Pi = 2*Integral_{x>=0} sin(x)/x dx. (End)
%F 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 Pi = Integral_{x = 0..2} sqrt(x/(2 - x)) dx. - _Arkadiusz Wesolowski_, Nov 20 2017
%F Pi = lim_{n->infinity} 2/n * Sum_{m=1,n} ( sqrt( (n+1)^2 - m^2 ) - sqrt( n^2 - m^2 ) ). - _Dimitri Papadopoulos_, May 31 2019
%F From _Peter Bala_, Oct 29 2019: (Start)
%F Pi = Sum_{n >= 0} 2^(n+1)/( binomial(2*n,n)*(2*n + 1) ) - Euler.
%F 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 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 Pi = Im(log(-i^i)) = log(i^i)*(-2). - _Peter Luschny_, Oct 29 2019
%F From _Amiram Eldar_, Aug 15 2020: (Start)
%F Equals 2 + Integral_{x=0..1} arccos(x)^2 dx.
%F Equals Integral_{x=0..oo} log(1 + 1/x^2) dx.
%F Equals Integral_{x=0..oo} log(1 + x^2)/x^2 dx.
%F Equals Integral_{x=-oo..oo} exp(x/2)/(exp(x) + 1) dx. (End)
%F Equals 4*(1/2)!^2 = 4*Gamma(3/2)^2. - _Gary W. Adamson_, Aug 23 2021
%F From _Peter Bala_, Dec 08 2021: (Start)
%F 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 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 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 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 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 For odd n, Pi = (2^(n-1)/A001818((n-1)/2))*gamma(n/2)^2. - _Alan Michael Gómez Calderón_, Mar 11 2022
%F 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 Pi = sqrt(3)*(27*S - 36)/24, where S = A248682. - _Peter Luschny_, Jul 22 2022
%F Equals Integral_{x=0..1} 1/sqrt(x-x^2) dx. - _Michal Paulovic_, Sep 24 2023
%F From _Peter Bala_, Oct 28 2023: (Start)
%F Pi = 48*Sum_{n >= 0} (-1)^n/((6*n + 1)*(6*n + 3)*(6*n + 5)).
%F 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 Pi = 16/5 - (288/5)*Sum_{n >= 0} (-1)^n * (6*n + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 9)).
%F 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 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 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 From _Peter Bala_, Nov 10 2023: (Start)
%F 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 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 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 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 For any integer n, Pi = (-1)^n * 4 * Sum_{k >= 0} 1/(4*k + 1 + 2*n) - 1/(4*k + 3 - 2*n). (End)
%e 3.1415926535897932384626433832795028841971693993751058209749445923078164062\
%e 862089986280348253421170679821480865132823066470938446095505822317253594081\
%e 284811174502841027019385211055596446229489549303819...
%p Digits := 110: Pi*10^104:
%p ListTools:-Reverse(convert(floor(%), base, 10)); # _Peter Luschny_, Oct 29 2019
%t RealDigits[ N[ Pi, 105]] [[1]]
%t 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 (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 (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 (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 (PARI) first(n)= default(realprecision, n+10); digits(floor(Pi*10^(n-1))) \\ _David A. Corneth_, Jun 21 2022
%o (Haskell) -- see link: Literate Programs
%o import Data.Char (digitToInt)
%o a000796 n = a000796_list (n + 1) !! (n + 1)
%o a000796_list len = map digitToInt $ show $ machin' `div` (10 ^ 10) where
%o machin' = 4 * (4 * arccot 5 unity - arccot 239 unity)
%o unity = 10 ^ (len + 10)
%o arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where
%o arccot' x unity summa xpow n sign
%o | term == 0 = summa
%o | otherwise = arccot'
%o x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign)
%o where term = xpow `div` n
%o -- _Reinhard Zumkeller_, Nov 24 2012
%o (Haskell) -- See Niemeijer link and also Gibbons link.
%o a000796 n = a000796_list !! (n-1) :: Int
%o a000796_list = map fromInteger $ piStream (1, 0, 1)
%o [(n, a*d, d) | (n, d, a) <- map (\k -> (k, 2 * k + 1, 2)) [1..]] where
%o piStream z xs'@(x:xs)
%o | lb /= approx z 4 = piStream (mult z x) xs
%o | otherwise = lb : piStream (mult (10, -10 * lb, 1) z) xs'
%o where lb = approx z 3
%o approx (a, b, c) n = div (a * n + b) c
%o mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f)
%o -- _Reinhard Zumkeller_, Jul 14 2013, Jun 12 2013
%o (Magma) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // _Bruno Berselli_, Mar 12 2013
%o (Python) from sympy import pi, N; print(N(pi, 1000)) # _David Radcliffe_, Apr 10 2019
%o (Python)
%o from mpmath import mp
%o def A000796(n):
%o if n >= len(A000796.str): mp.dps = n*6//5+50; A000796.str = str(mp.pi-5/mp.mpf(10)**mp.dps)
%o return int(A000796.str[n if n>1 else 0])
%o A000796.str = '' # _M. F. Hasler_, Jun 21 2022
%o (SageMath)
%o m=125
%o x=numerical_approx(pi, digits=m+5)
%o a=[ZZ(i) for i in x.str(skip_zeroes=True) if i.isdigit()]
%o a[:m] # _G. C. Greubel_, Jul 18 2023
%Y Cf. A001203 (continued fraction).
%Y 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 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 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 Cf. A248682.
%K cons,nonn,nice,core,easy
%O 1,1
%A _N. J. A. Sloane_
%E Additional comments from _William Rex Marshall_, Apr 20 2001
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