1,1

Sometimes called Archimedes's constant.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

J. Arndt & C. Haenel, Pi Unleashed, Springer NY 2001.

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]

P. Beckmann, A History of Pi, Golem Press, Boulder, CO, 1977.

J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997.

P. Eyard and J.-P. Lafon, The Number Pi, Amer. Math. Soc., 2004.

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.4.

Le Petit Archimede, Special Issue On Pi, Supplement to No. 64-5, May 1980 ADCS Amiens.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 31.

D. Shanks and J. W. Wrench. Jr., Calculation of pi to 100,000 decimals. Math. Comp. 16 1962 76-99.

J. Sondow, A faster product for Pi and a new integral for ln Pi/2, Amer. Math. Monthly 112 (2005) 729-734.

Harry J. Smith, Table of n, a(n) for n=1,...,20000

Dave Andersen, Pi-Search Page

Anonymous, A million digits of Pi

Anonymous, Liste de quelques milliers de decimales du nombre de pi

D. H. Bailey, On Kanada's computation of 1.24 trillion digits of Pi

D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications

J. M. Borwein, Talking about Pi

J. M. Borwein and M. Macklem, The (Digital) Life of Pi

J. Britton, Mnemonics For The Number Pi

J. P. Chabert, Pi up to 2000 decimals

E. S. Croot, Pade Approximations and the Transcendence of pi

L. Euler, On the sums of series of reciprocals

L. Euler, De summis serierum reciprocarum, E41.

Eureka, Tout pi or not tout pi

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants

GJ, 10 million digits of Pi

X. Gourdon, Pi to 16000 decimals

Xavier Gourdon, A new algorithm for computing Pi in base 10

B. Gourevitch, L'univers de Pi

L. Grebelius, Approximation of Pi: First 1000000 digits

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent

H. Havermann, Simple Continued Fraction for Pi

M. D. Huberty et al., 100000 Digits of Pi

ICON Project, Pi to 50000 places

P. Johns, 120000 Digits of Pi

Kanada Laboratory, 1.24 trillion digits of Pi

Yasumasa Kanada and Daisuke Takahashi, 206 billion digits of Pi

J. Moyer, First 10000 digits of pi

NERSC, Search Pi

Steve Pagliarulo, Stu's pi page ...

I. Peterson, A Passion for Pi

G. M. Phillips, Table of contents of "Pi: A source Book"

S. Plouffe, Plouffe's Inverter, 10000 digits of Pi

D. Pothet, Chronologie du calcul des decimales de pi

S. Ramanujan, Modular equations and approximations to \pi, Quart. J. Math. 45 (1914), 350-372.

H. Ricardo, Review of "The Number Pi" by P. Eymard & J.-P. Lafon

Daniel Sedory, The Pi Pages

Sizes, pi

A. Sofo, Pi and some other constants, J. Inequ. Pure Appl. Math. 6 (2005) vol. 5, #138

J. Sondow, A faster product for Pi and a new integral for ln Pi/2

D. Surendran, Can I have a small container of coffee?

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wikipedia, Pi

Index entries for sequences related to the number Pi

Jean-Louis Sigrist, Les 128000 premieres decimales du nombre PI [From Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 28 2009]

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]

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]

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]

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]

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]

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]

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]

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]

Pi=2*n*A000111(n-1)/A000111(n) as n-->infinity (conjecture). [From Mats Granvik (mats.granvik(AT)abo.fi), Aug 12 2009]

3.1415926535897932384626433832795028841971693993751058209749445923078164062\

862089986280348253421170679821480865132823066470938446095505822317253594081\

284811174502841027019385211055596446229489549303820...

RealDigits[ N[ Pi, 105]] [[1]]

(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 */

(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]

Pi in various bases: A004601 to A004608, A000796, A068436 to A068440, A062964. Cf. A007514.

Cf. A092798, A122214.

Cf. A133766,A133767. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Jan 11 2009]

Sequence in context: A013705 A087478 A112602 this_sequence A114609 A068089 A068079

Adjacent sequences: A000793 A000794 A000795 this_sequence A000797 A000798 A000799

cons,nonn,nice,core

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from William Rex Marshall (w.r.marshall(AT)actrix.co.nz), Apr 20, 2001

Corrected broken URL R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 31 2009