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A000796
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Decimal expansion of Pi.
(Formerly M2218 N0880)
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443
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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;
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OFFSET
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1,1
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COMMENTS
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Sometimes called Archimedes's constant.
Ratio of a circle's circumference to its diameter.
Also area of a circle with radius 1.
Also surface area of a sphere with diameter 1.
Also ratio of surface areas between a sphere and one of the faces of the circumscribed cube. Also ratio of volumes between a sphere and one of the six inscribed pyramids in the circumscribed cube. - Omar E. Pol, Aug 09 2012
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REFERENCES
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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.
Dario Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163. [N. J. A. Sloane, Mar 24 2012]
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
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Harry J. Smith, Table of n, a(n) for n=1,...,20000
Dave Andersen, Pi-Search Page
Anonymous, A million digits of Pi [broken link]
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 [broken link]
D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the AMS, Volume 52, Number 5, May 2005, pp. 502-514.
Frits Beukers, A rational approach to Pi, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.
J. M. Borwein, Talking about Pi
J. M. Borwein and M. Macklem, The (Digital) Life of Pi, The Australian Mathematical Society Gazette, Volume 33, Number 5, Sept. 2006, pp. 243-248.
Peter Borwein, The amazing number Pi, Nieuw Archief voor de Wiskunde, September 2000, pp. 254-258.
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 [broken link]
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 [broken link]
Yasumasa Kanada and Daisuke Takahashi, 206 billion digits of Pi [broken link]
Literate Programs, Pi with Machin's formula (Haskell)
Johannes W. Meijer, Pi everywhere poster, Mar 14 2013
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"
_Simon Plouffe_, Plouffe's Inverter, 10000 digits of Pi [broken link]
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 [broken link]
M. Ripa and G. Morelli, Retro-analytical Reasoning IQ tests for the High Range, 2013.
Daniel B. Sedory, The Pi Pages
Jean-Louis Sigrist, Les 128000 premieres decimales du nombre PI [From Lekraj Beedassy, Sep 28 2009]
Sizes, pi
A. Sofo, Pi and some other constants, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, Issue 5, Article 138, 2005.
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? [broken link]
Wislawa Szymborska, Pi (The admirable number Pi), Miracle Fair, 2002.
Stan Wagon, Is Pi Normal?
Eric Weisstein's World of Mathematics, Pi
Wikipedia, Pi
Wikipedia, Normal Number
Wikipedia, Bailey-Borwein-Plouffe formula
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
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Pi = 4 * Sum_{ k >= 0} (-1)^k/(2k+1) [Madhava-Gregory-Leibniz, 1450-1671] - N. J. A. Sloane, Feb 27 2013
From Johannes W. Meijer, Mar 10 2013: (Start)
2/Pi = (sqrt(2)/2) * (sqrt(2 + sqrt(2))/2) * (sqrt(2 + sqrt(2 + sqrt(2)))/2) * … [Viete, 1593]
2/Pi = product(k >= 1, product((4*k^2-1)/(4*k^2)) [Wallis, 1655]
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]
Pi/4 = 4*arctan(1/5) - arctan(1/239) [Machin, 1706]
Pi^2/6 = 3*sum(n >= 1, 1/(n^2*binomial(2*n,n))) [Euler, 1748]
1/Pi = (2*sqrt(2)/9801) * sum(n >= 0, (4*n)!*(1103+26390*n)/((n!)^4*396^(4*n))) [Ramanujan, 1914]
1/Pi = 12*sum((-1)^n*(6*n)!/((n!)^3*(3*n)!)*((13591409 + 545140134*n)/(64032^3)(n+1/2)) [David and Gregory Chudnovsky, 1989]
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)
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, 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, 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, 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, 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, Dec 25 2008]
Pi = 4*Sum(1/(4*k+1) - 1/(4*k+3),k = 0 .. infinity) [From Alexander R. Povolotsky, 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, Jan 11 2009]
Pi=4*sqrt(-1*(sum((I^(2*n+1))/(2*n+1),n=0...infinity)^2)) [From Alexander R. Povolotsky, Jan 25 2009]
Pi=2*n*A000111(n-1)/A000111(n) as n-->infinity (conjecture). [From Mats Granvik, Aug 12 2009]
Pi = integral_{x=-Infinity..Infinity} dx/(1+x^2). [From Mats Granvik & Gary W. Adamson, Sep 23 2012]
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EXAMPLE
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3.1415926535897932384626433832795028841971693993751058209749445923078164062\
862089986280348253421170679821480865132823066470938446095505822317253594081\
284811174502841027019385211055596446229489549303820...
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MATHEMATICA
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RealDigits[ N[ Pi, 105]] [[1]]
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PROG
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(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)); } [Harry J. Smith, Apr 15 2009]
(Haskell) see link: Literate Programs
import Data.Char (digitToInt)
a000796 n = a000796_list (n + 1) !! (n + 1)
a000796_list len = map digitToInt $ show $ machin' `div` (10 ^ 10) where
machin' = 4 * (4 * arccot 5 unity - arccot 239 unity)
unity = 10 ^ (len + 10)
arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where
arccot' x unity summa xpow n sign
| term == 0 = summa
| otherwise = arccot'
x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign)
where term = xpow `div` n
-- Reinhard Zumkeller, Nov 24 2012
(MAGMA) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // Bruno Berselli, Mar 12 2013
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CROSSREFS
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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), A224750 (b=26), A224751 (b=27), A060707 (b=60). - Jason Kimberley, Dec 06 2012
Cf. A003881 (Pi/4), A007514 (Pi), A092798 (Pi/2), A122214 (Pi/2), A133766, A133767.
Cf. A002388 (pi^2), A091925 (pi^3), A092425 (pi^4), A092731 (pi^5), A092732 (pi^6), A092735 (pi^7), A092736 (pi^8).
Cf. A001901 (Pi/2; Wallis), A054387 (Pi; Newton), A096954 (Pi/4; Machin), A013661 (Pi^2/6; Euler), A019727 (sqrt(2*Pi)), A059956 (6/Pi^2), A002736 (Pi^2/18), A048581 (Pi; BBP), A097486 (Pi), A019692 (2*Pi; tau), A060294 (2/Pi), A163973 (Pi/ln(2)), A166107 (Pi; MGL).
Sequence in context: A216548 A087478 A112602 * A212131 A114609 A068089
Adjacent sequences: A000793 A000794 A000795 * A000797 A000798 A000799
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KEYWORD
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cons,nonn,nice,core,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Additional comments from William Rex Marshall, Apr 20, 2001
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STATUS
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approved
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