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A000796
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Decimal expansion of Pi.
(Formerly M2218 N0880)
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388
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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
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Sometimes called Archimedes's constant.
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REFERENCES
| Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.
Mohammad K. Azarian, The Introduction of Al-Risala al-Muhitiyya: An English Translation, International Journal of Pure and Applied Mathematics, Vol. 57, No. 6, 2009 , pp. 903-914.
Mohammad K. Azarian, Al-Kashi's Fundamental Theorem, International Journal of Pure and Applied Mathematics, Vol. 14, No. 4, 2004, pp. 499-509. Mathematical Reviews, MR2005b:01021 (01A30), February 2005, p. 919. Zentralblatt MATH, Zbl 1059.01005.
Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
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.
J. Arndt & C. Haenel, Pi Unleashed, Springer NY 2001.
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.
J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270.
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Sondow, A faster product for Pi and a new integral for ln Pi/2, Amer. Math. Monthly 112 (2005) 729-734.
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LINKS
| 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 B. Sedory, The Pi Pages
Jean-Louis Sigrist, Les 128000 premieres decimales du nombre PI [From Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 28 2009]
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, Pi
Wikipedia, Bailey-Borwein-Plouffe formula
Wikipedia, Pi
Alexander J. Yee & Shigeru Kondo, 5 Trillion Digits of Pi - New World Record
Alexander J. Yee & Shigeru Kondo, Round 2... 10 Trillon Digits of Pi
Index entries for sequences related to the number Pi
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FORMULA
| 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]
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EXAMPLE
| 3.1415926535897932384626433832795028841971693993751058209749445923078164062\
862089986280348253421170679821480865132823066470938446095505822317253594081\
284811174502841027019385211055596446229489549303820...
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MATHEMATICA
| RealDigits[ N[ Pi, 105]] [[1]]
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PROG
| (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]
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CROSSREFS
| Cf. A001203 (continued fraction).
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).
Cf. A007514.
Cf. A092798, A122214.
Cf. A133766, A133767. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Jan 11 2009]
Cf. A002388 (pi^2), A091925 (pi^3), A092425 (pi^4), A092731 (pi^5), A092732 (pi^6), A092735 (pi^7), A092736 (pi^8).
Sequence in context: A013705 A087478 A112602 * A114609 A068089 A068079
Adjacent sequences: A000793 A000794 A000795 * A000797 A000798 A000799
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KEYWORD
| cons,nonn,nice,core
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| 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
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