1,1

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)

Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.

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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).

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

Dave Andersen, Pi-Search Page

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J. M. Borwein, Talking about Pi

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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

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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

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, arXiv:math/0506319 [math.NT], 2005-2006.

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Index entries for sequences related to the number Pi

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

3.1415926535897932384626433832795028841971693993751058209749445923078164062\

862089986280348253421170679821480865132823066470938446095505822317253594081\

284811174502841027019385211055596446229489549303820...

Digits:=1000; evalf(Pi); # Wesley Ivan Hurt, Oct 24 2013

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(%))) /* 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

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/log(2)), A166107 (Pi; MGL).

See A245770 for an interesting sieve related to this sequence.

Sequence in context: A247385 A253214 A112602 * A212131 A114609 A068089

Adjacent sequences:  A000793 A000794 A000795 * A000797 A000798 A000799

cons,nonn,nice,core,easy,changed

N. J. A. Sloane

Additional comments from William Rex Marshall, Apr 20 2001

approved