login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000796 Decimal expansion of Pi (or, digits of Pi).
(Formerly M2218 N0880)
547
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.

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 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

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.

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.

LINKS

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.

Christian Boyer, MultiMagic Squares

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

Jeremy Gibbons, , Unbounded Spigot Algorithms for the Digits of Pi

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

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

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

Eric Weisstein's World of Mathematics, Pi Digits

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

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, 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)

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 & Gary W. Adamson, Sep 23 2012]

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(%))) /* 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

(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

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).

See A245770 for an interesting sieve related to this sequence.

Sequence in context: A087478 A247385 A112602 * A212131 A114609 A068089

Adjacent sequences:  A000793 A000794 A000795 * A000797 A000798 A000799

KEYWORD

cons,nonn,nice,core,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Additional comments from William Rex Marshall, Apr 20, 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified September 30 05:01 EDT 2014. Contains 247417 sequences.