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 A000796 Decimal expansion of Pi (or, digits of Pi). (Formerly M2218 N0880) 625
 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; text; internal format)
 OFFSET 1,1 COMMENTS 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. A useful mnemonic for remembering the first few terms: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics ... 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 Also surface area of a quarter of a sphere of radius 1. - Omar E. Pol, Oct 03 2013 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..infinity} (-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 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] x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 25 2013 Also diameter of a sphere whose surface area equals the volume of the circumscribed cube. - Omar E. Pol, Jan 13 2014 From Daniel Forgues, Mar 20 2015: (Start) An interesting anecdote about the base 10 representation of Pi, with 3 (integer part) as first (index 1) digit:   358 0   359 3   360 6   361 0   362 0 And the circle is customarily subdivided into 360 degrees (although Pi radians yields half the circle)... (End) 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. 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. 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. LINKS Harry J. Smith, Table of n, a(n) for n = 1..20000 Dave Andersen, Pi-Search Page Anonymous, A million digits of 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. Christian Boyer, MultiMagic Squares J. Britton, Mnemonics For The Number Pi D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163. 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, arXiv:math/0506415 [math.HO], 2005-2008. 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 Jeremy Gibbons, Unbounded Spigot Algorithms for the 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, Ramanujan J. 16 (2008) 247-270. Preprint: arXiv:math/0506319 [math.NT] (2005-2006). 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 Remco Niemeijer, , The Digits of Pi, programmingpraxis 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, 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 [broken link] M. Ripa and G. Morelli, Retro-analytical Reasoning IQ tests for the High Range,  2013. Daniel B. Sedory, The Pi Pages D. Shanks and J. W. Wrench. Jr., Calculation of pi to 100,000 decimals, Math. Comp. 16 1962 76-99. Jean-Louis Sigrist, Les 128000 premieres decimales du nombre PI 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, arXiv:math/0401406 [math.NT], 2004. D. Surendran, Can I have a small container of coffee? [broken link] Wislawa Szymborska, Pi (The admirable number Pi), Miracle Fair, 2002. Jonas Castillo Toloza, Fascinating Method for Finding Pi G. Vacca, A new analytical expression for the number pi, and some historical considerations, Bull. Amer. Math. Soc. 16 (1910), 368-369. Stan Wagon, Is Pi Normal? Eric Weisstein's World of Mathematics, Pi and Pi Digits Wikipedia, Bailey-Borwein-Plouffe formula, Normal Number, and 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 FORMULA 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} (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_{n>=0} (-1)^n*(6*n)!*(13591409 + 545140134*n)/((3*n)!*(n!)^3*(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) Pi = 4 * Sum_{k>=0} 1/(4*k+1) - 1/(4*k+3). - Alexander R. Povolotsky, Dec 25 2008 Pi = 4*sqrt(-1*(Sum_{n>=0} (i^(2*n+1))/(2*n+1))^2). - Alexander R. Povolotsky, Jan 25 2009 Pi = 2*n*A000111(n-1)/A000111(n) as n-->infinity (conjecture). - Mats Granvik, Aug 12 2009 Pi = Integral_{x=-infinity..infinity} dx/(1+x^2). - Mats Granvik and Gary W. Adamson, Sep 23 2012 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 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] EXAMPLE 3.1415926535897932384626433832795028841971693993751058209749445923078164062\ 862089986280348253421170679821480865132823066470938446095505822317253594081\ 284811174502841027019385211055596446229489549303820... MAPLE Digits:=1000; evalf(Pi); # Wesley Ivan Hurt, Oct 24 2013 MATHEMATICA RealDigits[ N[ Pi, 105]] [[1]] 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(%))) /* Bill 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 (Haskell) See Niemeijer link and also Gibbons link. a000796 n = a000796_list !! (n-1) :: Int a000796_list = map fromInteger \$ piStream (1, 0, 1)    [(n, a*d, d) | (n, d, a) <- map (\k -> (k, 2 * k + 1, 2)) [1..]] where    piStream z xs'@(x:xs)      | lb /= approx z 4 = piStream (mult z x) xs      | otherwise = lb : piStream (mult (10, -10 * lb, 1) z) xs'      where lb = approx z 3            approx (a, b, c) n = div (a * n + b) c            mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f) -- Reinhard Zumkeller, Jul 14 2013, Jun 12 2013 (MAGMA) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // Bruno Berselli, Mar 12 2013 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), A224750 (b=26), A224751 (b=27), A060707 (b=60). - Jason Kimberley, Dec 06 2012 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)). 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). See A245770 for an interesting sieve related to this sequence. Sequence in context: A247385 A253214 A112602 * A212131 A114609 A271452 Adjacent sequences:  A000793 A000794 A000795 * A000797 A000798 A000799 KEYWORD cons,nonn,nice,core,easy,changed AUTHOR EXTENSIONS Additional comments from William Rex Marshall, Apr 20 2001 STATUS approved

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