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%I A000796 M2218 N0880
%S A000796 3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,8,
%T A000796 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,
%U A000796 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.
%C A000796 Sometimes called Archimedes's constant.
%D A000796 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000796 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000796 J. Arndt & C. Haenel, Pi Unleashed, Springer NY 2001.
%D A000796 Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics 
               Journal, Vol. 39, No. 1, 2008, pp. 66. Solution appeared in Vol. 
               40, No. 1, 2009, pp. 62-64. [From Mohammad K. Azarian (azarian(AT)evansville.edu), 
               Feb 08 2009]
%D A000796 P. Beckmann, A History of Pi, Golem Press, Boulder, CO, 1977.
%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 D. Shanks and J. W. Wrench. Jr., Calculation of pi to 100,000 decimals. 
               Math. Comp. 16 1962 76-99.
%D A000796 J. Sondow, A faster product for Pi and a new integral for ln Pi/2, Amer. 
               Math. Monthly 112 (2005) 729-734.
%H A000796 Harry J. Smith, <a href="b000796.txt">Table of n, a(n) for n=1,...,20000</
               a>
%H A000796 Dave Andersen, <a href="http://www.angio.net/pi/piquery">Pi-Search Page</
               a>
%H A000796 Anonymous, <a href="http://www.exploratorium.edu/pi/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="http://www.nersc.gov/~dhbailey/dhbpapers/dhb-kanada.pdf">
               On Kanada's computation of 1.24 trillion digits of Pi</a>
%H A000796 D. H. Bailey and J. M. Borwein, <a href="http://eprints.cecm.sfu.ca/archive/
               00000269/">Experimental Mathematics: Examples, Methods and Implications</
               a>
%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://eprints.cecm.sfu.ca/archive/
               00000263/">The (Digital) Life of Pi</a>
%H A000796 J. Britton, <a href="http://britton.disted.camosun.bc.ca/jbpimem.htm">
               Mnemonics For The Number Pi</a>
%H A000796 J. P. Chabert, <a href="http://jpm-chabert.club.fr/maths/Pi.html">Pi 
               up to 2000 decimals</a>
%H A000796 E. S. Croot, <a href="http://www.math.gatech.edu/~ecroot/transcend.pdf">
               Pade Approximations and the Transcendence of pi</a>
%H A000796 L. Euler, <a href="http://arXiv.org/abs/math.HO/0506415">On the sums 
               of series of reciprocals</a>
%H A000796 L. Euler, <a href="http://www.eulerarchive.org">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 GJ, <a href="http://gj.mit.edu/pi/digits/10million.txt">10 million digits 
               of Pi</a>
%H A000796 X. Gourdon, <a href="http://webs.adam.es/rllorens/pi.htm">Pi to 16000 
               decimals</a>
%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 B. Gourevitch, <a href="http://www.pi314.net">L'univers de Pi</a>
%H A000796 L. Grebelius, <a href="http://www2.tripnet.se/~nlg/pi0001.htm">Approximation 
               of Pi: First 1000000 digits</a>
%H A000796 J. Guillera and J. Sondow, <a href="http://arXiv.org/abs/math.NT/0506319">
               Double integrals and infinite products for some classical constants 
               via analytic continuations of Lerch's transcendent</a>
%H A000796 H. Havermann, <a href="http://chesswanks.com/pxp/cfpi.html">Simple Continued 
               Fraction for Pi</a>
%H A000796 M. D. Huberty et al., <a href="http://www.geom.umn.edu/~huberty/math5337/
               groupe/digits.html">100000 Digits of Pi</a>
%H A000796 ICON Project, <a href="http://www.cs.arizona.edu/icon/oddsends/pi.htm">
               Pi to 50000 places</a>
%H A000796 P. Johns, <a href="http://www.wpdpi.com/pi.shtml">120000 Digits of Pi</
               a>
%H A000796 Kanada Laboratory, <a href="http://pi2.cc.u-tokyo.ac.jp/index.html">1.24 
               trillion digits of Pi</a>
%H A000796 Yasumasa Kanada and Daisuke Takahashi, <a href="http://www.cecm.sfu.ca/
               personal/jborwein/Kanada_200b.html">206 billion digits of Pi</a>
%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>
%H A000796 Steve Pagliarulo, <a href="http://home.istar.ca/~lyster/pi.html">Stu's 
               pi page</a> ...
%H A000796 I. Peterson, <a href="http://www.maa.org/mathland/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 S. Plouffe, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/
               pi10000.txt">10000 digits of Pi</a>
%H A000796 D. Pothet, <a href="http://perso.wanadoo.fr/didier.pothet/pi.html">Chronologie 
               du calcul des decimales de pi</a>
%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/reviews/thenumberpi.html">Review 
               of "The Number Pi" by P. Eymard & J.-P. Lafon</a>
%H A000796 Daniel Sedory, <a href="http://www.geocities.com/tsrmath/pi/index.html">
               The Pi Pages</a>
%H A000796 Sizes, <a href="http://www.sizes.com/numbers/pi.htm">pi</a>
%H A000796 A. Sofo, <a href="http://jipam.vu.edu.au/article.php?sid=613">Pi and 
               some other constants</a>, J. Inequ. Pure Appl. Math. 6 (2005) vol. 
               5, #138
%H A000796 J. Sondow, <a href="http://arXiv.org/abs/math.NT/0401406">A faster product 
               for Pi and a new integral for ln Pi/2</a>
%H A000796 D. Surendran, <a href="http://www.uz.ac.zw/science/maths/zimaths/pimnem.htm">
               Can I have a small container of coffee?</a>
%H A000796 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Pi.html">Link to a section of The World of Mathematics.</a>
%H A000796 Wikipedia, <a href="http://www.wikipedia.org/wiki/Pi">Pi</a>
%H A000796 <a href="Sindx_Ph.html#Pi314">Index entries for sequences related to 
               the number Pi</a>
%H A000796 Jean-Louis Sigrist, <a href="http://jlsigrist.com/pi.html">Les 128000 
               premieres decimales du nombre PI</a> [From Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Sep 28 2009]
%F A000796 Alexander R. Povolotsky came up with the following BBP-type formula: 
               Pi= 2/3 * (-1 + Sum(7/(4*k+1) - 6/(4*k+3) - 1/(4*k+5),k = 0 .. infinity). 
               J. Guillera noted: "There is an easy proof of that formula if to 
               convert it into an integral. In doing the proof, observe that int_(0,
               1) x ^ (4n+a) = 1 / (4n+a+1). The proof is easy but it can be interesting 
               if one does not know the method. The formula converges slowly because 
               there is not a factor like for example 1/16^k." Roger Bagula tried 
               to use this formula for generating high quality pseudo-random level 
               results. He thinks that this formula's algorithm is faster and takes 
               less computer memory for that (comparing with regular BBP) [From 
               Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 30 2008]
%F A000796 Pi = 2/3 * (-1 + Sum(7/(4*k+1) - 6/(4*k+3) - 1/(4*k+5),k = 0 .. infinity) 
               [From Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 30 2008]
%F A000796 Another (ugly) formula for Pi (in Maple syntax): Pi = 6/7*(1/3*sum((843*n 
               + 4607)/((n+5)*(3*n+7)*(3*n+22)),n=0...infinity) - 655999/248976 
               - 7/2*ln(3))*sqrt(3) [From Alexander R. Povolotsky (pevnev(AT)juno.com), 
               Dec 07 2008]
%F A000796 Pi = (4/5)*(Sum(7/(4*k+1) - 5/(4*k+3) - 2/(4*k+5),k = 0 .. infinity) 
               -2) [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 25 2008]
%F A000796 Pi = Sum(7/(4*k+1) - 4/(4*k+3) - 3/(4*k+5),k = 0 .. infinity) - 3 [From 
               Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 25 2008]
%F A000796 Pi = 4*Sum(1/(4*k+1) - 1/(4*k+3),k = 0 .. infinity) [From Alexander R. 
               Povolotsky (pevnev(AT)juno.com), Dec 25 2008]
%F A000796 pi = c + sum( k>=0, (4-c)/(4k+1) -4/(4k+3) +c/(4k+5) ) for any c. [From 
               Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Jan 11 2009]
%F A000796 Pi=4*sqrt(-1*(sum((I^(2*n+1))/(2*n+1),n=0...infinity)^2)) [From Alexander 
               R. Povolotsky (pevnev(AT)juno.com), Jan 25 2009]
%F A000796 Pi=2*n*A000111(n-1)/A000111(n) as n-->infinity (conjecture). [From Mats 
               Granvik (mats.granvik(AT)abo.fi), Aug 12 2009]
%e A000796 3.1415926535897932384626433832795028841971693993751058209749445923078164062\
%e A000796 862089986280348253421170679821480865132823066470938446095505822317253594081\
%e A000796 284811174502841027019385211055596446229489549303820...
%t A000796 RealDigits[ N[ Pi, 105]] [[1]]
%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(%))) /* 
               R. W. 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)); } [From Harry J. Smith 
               (hjsmithh(AT)sbcglobal.net), Apr 15 2009]
%Y A000796 Pi in various bases: A004601 to A004608, A000796, A068436 to A068440, 
               A062964. Cf. A007514.
%Y A000796 Cf. A092798, A122214.
%Y A000796 Cf. A133766,A133767. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Jan 11 2009]
%Y A000796 Sequence in context: A013705 A087478 A112602 this_sequence A114609 A068089 
               A068079
%Y A000796 Adjacent sequences: A000793 A000794 A000795 this_sequence A000797 A000798 
               A000799
%K A000796 cons,nonn,nice,core
%O A000796 1,1
%A A000796 N. J. A. Sloane (njas(AT)research.att.com).
%E A000796 Additional comments from William Rex Marshall (w.r.marshall(AT)actrix.co.nz), 
               Apr 20, 2001
%E A000796 Corrected broken URL R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 
               31 2009

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