A000796 Decimal expansion of Pi (or digits of Pi).
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
Offset: 1
Examples
3.1415926535897932384626433832795028841971693993751058209749445923078164062\ 862089986280348253421170679821480865132823066470938446095505822317253594081\ 284811174502841027019385211055596446229489549303819...
References
- 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.
- Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 396.
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 237-239.
- 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).
- Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equations 1:7:1, 1:7:2 at pages 12-13.
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See pp. 48-55.
Links
- Harry J. Smith, Table of n, a(n) for n = 1..20000
- Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, and Brendan M. Quine, Unconditional applicability of the Lehmer's measure to the two-term Machin-like formula for pi, arXiv:2004.11711 [math.GM], 2020.
- Emilio Ambrisi and Bruno Rizzi, Appunti da un corso di aggiornamento, Mathesis (Sezione Casertana), Quaderno n. 1, Liceo G. Galilei, Mondragone (CE), Italy, June 22-28 1979. (In Italian). See p. 15.
- 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 [archived page]
- 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.
- Harry Baker, "Pi calculated to a record-breaking 62.8 trillion digits", Live Science, August 17, 2021.
- Steve Baker and Thomas Moore, 100 trillion digits of pi
- 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 [archived page]
- D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163.
- Jonas Castillo Toloza, Fascinating Method for Finding Pi
- 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 [archived page]
- Xavier Gourdon, A new algorithm for computing Pi in base 10
- X. Gourdon and P. Sebah, Archimedes' constant Pi
- B. Gourevitch, L'univers de Pi
- Antonio Gracia Llorente, Novel Infinite Products πe and π/e, OSF Preprint, 2024.
- 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).
- Carl-Johan Haster, Pi from the sky -- A null test of general relativity from a population of gravitational wave observations, arXiv:2005.05472 [gr-qc], 2020.
- H. Havermann, Simple Continued Fraction for Pi [archived page]
- M. D. Huberty et al., 100000 Digits of Pi
- ICON Project, Pi to 50000 places [archived page]
- Emma Haruka Iwao, Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes' constant on Google Cloud
- P. Johns, 120000 Digits of Pi [archived page]
- Yasumasa Kanada, 1.24 trillion digits of Pi
- Yasumasa Kanada and Daisuke Takahashi, 206 billion digits of Pi [archived page]
- Literate Programs, Pi with Machin's formula (Haskell) [archived page]
- Johannes W. Meijer, Pi everywhere poster, Mar 14 2013
- J. Moyer, First 10000 digits of pi
- NERSC, Search Pi [broken link]
- Remco Niemeijer, The Digits of Pi, programmingpraxis.
- Steve Pagliarulo, Stu's pi page [archived page]
- Chittaranjan Pardeshi, BBP-Like formula for Pi in Golden Ratio Base Phi
- Michael Penn, A nice inverse tangent integral., YouTube video, 2020.
- Michael Penn, Pi is irrational (π∉ℚ), YouTube video, 2020.
- I. Peterson, A Passion for Pi
- G. M. Phillips, Table of contents of "Pi: A source Book"
- Simon Plouffe, 10000 digits of Pi
- Simon Plouffe, A formula for the nth decimal digit or binary of Pi and powers of Pi, arXiv:2201.12601 [math.NT], 2022.
- D. Pothet, Chronologie du calcul des decimales de pi [broken link]
- M. Z. Rafat and D. Dobie, Throwing Pi at a wall, arXiv:1901.06260 [physics.class-ph], 2020.
- 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
- M. Ripa and G. Morelli, Retro-analytical Reasoning IQ tests for the High Range, 2013.
- Grant Sanderson, Why do colliding blocks compute pi?, 3Blue1Brown video (2019).
- Daniel B. Sedory, The Pi Pages
- Michael I. Shamos, A catalog of the real numbers, (2007). See p. 180.
- 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
- N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
- Katie Steckles, Pi and constrained writing, The Aperiodical, 2015.
- A. Sofo, Pi and some other constants, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, Issue 5, Article 138, 2005.
- Jonathan Sondow, A faster product for Pi and a new integral for ln Pi/2, arXiv:math/0401406 [math.NT], 2004; Amer. Math. Monthly 112 (2005) 729-734.
- D. Surendran, Can I have a small container of coffee? [archived page]
- Wislawa Szymborska, Pi (The admirable number Pi), Miracle Fair, 2002.
- 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, Pi, and Machin-like formula
- Wikipedia, Pilish.
- Alexander J. Yee & Shigeru Kondo, 5 Trillion Digits of Pi - New World Record
- Alexander J. Yee & Shigeru Kondo, Round 2... 10 Trillion Digits of Pi
- Index entries for sequences related to the number Pi
- Index entries for "core" sequences
- Index entries for transcendental numbers
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)).
Programs
-
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 -
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 */
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Magma
pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // Bruno Berselli, Mar 12 2013
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Maple
Digits := 110: Pi*10^104: ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Oct 29 2019
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Mathematica
RealDigits[ N[ Pi, 105]] [[1]] Table[ResourceFunction["NthDigit"][Pi, n], {n, 1, 102}] (* Joan Ludevid, Jun 22 2022; easy to compute a(10000000)=7 with this function; requires Mathematica 12.0+ *) -
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 -
PARI
A796=[]; A000796(n)={if(n>#A796, localprec(n*6\5+29); A796=digits(Pi\.1^(precision(Pi)-3))); A796[n]} \\ NOTE: as the other programs, this returns the n-th term of the sequence, with n = 1, 2, 3, ... and not n = 1, 0, -1, -2, .... - M. F. Hasler, Jun 21 2022
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PARI
first(n)= default(realprecision, n+10); digits(floor(Pi*10^(n-1))) \\ David A. Corneth, Jun 21 2022
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PARI
lista(nn, p=20)= {my(u=10^(nn+p+1), f(x, u)=my(n=1, q=u\x, r=q, s=1, t); while(t=(q\=(x*x))\(n+=2), r+=(s=-s)*t); r*4); digits((4*f(5, u)-f(239, u))\10^(p+2)); } \\ Machin-like, with p > the maximal number of consecutive 9-digits to be expected (A048940) - Ruud H.G. van Tol, Dec 26 2024 -
Python
from sympy import pi, N; print(N(pi, 1000)) # David Radcliffe, Apr 10 2019
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Python
from mpmath import mp def A000796(n): if n >= len(A000796.str): mp.dps = n*6//5+50; A000796.str = str(mp.pi-5/mp.mpf(10)**mp.dps) return int(A000796.str[n if n>1 else 0]) A000796.str = '' # M. F. Hasler, Jun 21 2022
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SageMath
m=125 x=numerical_approx(pi, digits=m+5) a=[ZZ(i) for i in x.str(skip_zeroes=True) if i.isdigit()] a[:m] # G. C. Greubel, Jul 18 2023
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*(640320^3)^(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 = Integral_{x=-oo..oo} 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
From Ilya Gutkovskiy, Aug 07 2016: (Start)
Pi = Sum_{k>=1} (3^k - 1)*zeta(k+1)/4^k.
Pi = 2*Product_{k>=2} sec(Pi/2^k).
Pi = 2*Integral_{x>=0} sin(x)/x dx. (End)
Pi = 2^{k + 1}*arctan(sqrt(2 - a_{k - 1})/a_k) at k >= 2, where a_k = sqrt(2 + a_{k - 1}) and a_1 = sqrt(2). - Sanjar Abrarov, Feb 07 2017
Pi = Integral_{x = 0..2} sqrt(x/(2 - x)) dx. - Arkadiusz Wesolowski, Nov 20 2017
Pi = lim_{n->oo} 2/n * Sum_{m=1,n} ( sqrt( (n+1)^2 - m^2 ) - sqrt( n^2 - m^2 ) ). - Dimitri Papadopoulos, May 31 2019
From Peter Bala, Oct 29 2019: (Start)
Pi = Sum_{n >= 0} 2^(n+1)/( binomial(2*n,n)*(2*n + 1) ) - Euler.
More generally, Pi = (4^x)*x!/(2*x)! * Sum_{n >= 0} 2^(n+1)*(n+x)!*(n+2*x)!/(2*n+2*x+1)! = 2*4^x*x!^2/(2*x+1)! * hypergeom([2*x+1,1], [x+3/2], 1/2), valid for complex x not in {-1,-3/2,-2,-5/2,...}. Here, x! is shorthand notation for the function Gamma(x+1). This identity may be proved using Gauss's second summation theorem.
Setting x = 3/4 and x = -1/4 (resp. x = 1/4 and x = -3/4) in the above identity leads to series representations for the constant A085565 (resp. A076390). (End)
Pi = Im(log(-i^i)) = log(i^i)*(-2). - Peter Luschny, Oct 29 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals 2 + Integral_{x=0..1} arccos(x)^2 dx.
Equals Integral_{x=0..oo} log(1 + 1/x^2) dx.
Equals Integral_{x=0..oo} log(1 + x^2)/x^2 dx.
Equals Integral_{x=-oo..oo} exp(x/2)/(exp(x) + 1) dx. (End)
Equals 4*(1/2)!^2 = 4*Gamma(3/2)^2. - Gary W. Adamson, Aug 23 2021
From Peter Bala, Dec 08 2021: (Start)
Pi = 32*Sum_{n >= 1} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9))= 384*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)*(4*n^2 - 9)*(4*n^2 - 25)).
More generally, it appears that for k = 1,2,3,..., Pi = 16*(2*k)!*Sum_{n >= 1} (-1)^(n+k+1)*n^2/((4*n^2 - 1)* ... *(4*n^2 - (2*k+1)^2)).
Pi = 32*Sum_{n >= 1} (-1)^(n+1)*n^2/(4*n^2 - 1)^2 = 768*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)^2*(4*n^2 - 9)^2).
More generally, it appears that for k = 0,1,2,..., Pi = 16*Catalan(k)*(2*k)!*(2*k+2)!*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)^2* ... *(4*n^2 - (2*k+1)^2)^2).
Pi = (2^8)*Sum_{n >= 1} (-1)^(n+1)*n^2/(4*n^2 - 1)^4 = (2^17)*(3^5)*Sum_{n >= 2} (-1)^n*n^2*(n^2 - 1)/((4*n^2 - 1)^4*(4*n^2 - 9)^4) = (2^27)*(3^5)*(5^5)* Sum_{n >= 3} (-1)^(n+1)*n^2*(n^2 - 1)*(n^2 - 4)/((4*n^2 - 1)^4*(4*n^2 - 9)^4*(4*n^2 - 25)^4). (End)
For odd n, Pi = (2^(n-1)/A001818((n-1)/2))*gamma(n/2)^2. - Alan Michael Gómez Calderón, Mar 11 2022
Pi = 4/phi + Sum_{n >= 0} (1/phi^(12*n)) * ( 8/((12*n+3)*phi^3) + 4/((12*n+5)*phi^5) - 4/((12*n+7)*phi^7) - 8/((12*n+9)*phi^9) - 4/((12*n+11)*phi^11) + 4/((12*n+13)*phi^13) ) where phi = (1+sqrt(5))/2. - Chittaranjan Pardeshi, May 16 2022
Pi = sqrt(3)*(27*S - 36)/24, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Integral_{x=0..1} 1/sqrt(x-x^2) dx. - Michal Paulovic, Sep 24 2023
From Peter Bala, Oct 28 2023: (Start)
Pi = 48*Sum_{n >= 0} (-1)^n/((6*n + 1)*(6*n + 3)*(6*n + 5)).
More generally, it appears that for k >= 0 we have Pi = A(k) + B(k)*Sum_{n >= 0} (-1)^n/((6*n + 1)*(6*n + 3)*...*(6*n + 6*k + 5)), where A(k) is a rational approximation to Pi and B(k) = (3 * 2^(3*k+3) * (3*k + 2)!) / (2^(3*k+1) - (-1)^k). The first few values of A(k) for k >= 0 are [0, 256/85, 65536/20955, 821559296/261636375, 6308233216/2008080987, 908209489444864/289093830828075, ...].
Pi = 16/5 - (288/5)*Sum_{n >= 0} (-1)^n * (6*n + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 9)).
More generally, it appears that for k >= 0 we have Pi = C(k) + D(k)*Sum_{n >= 0} (-1)^n* (6*n + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 6*k + 3)), where C(k) and D(k) are rational numbers. The case k = 0 is the Madhava-Gregory-Leibniz series for Pi.
Pi = 168/53 + (288/53)*Sum_{n >= 0} (-1)^n * (42*n^2 + 25*n)/((6*n + 1)*(6*n + 3)*(6*n + 5)*(6*n + 7)).
More generally, it appears that for k >= 1 we have Pi = E(k) + F(k)*Sum_{n >= 0} (-1)^n * (6*(6*k + 1)*n^2 + (24*k + 1)*n)/((6*n + 1)*(6*n + 3)*...*(6*n + 6*k + 1)), where E(k) and F(k) are rational numbers. (End)
From Peter Bala, Nov 10 2023: (Start)
The series representation Pi = 4 * Sum_{k >= 0} 1/(4*k + 1) - 1/(4*k + 3) given above by Alexander R. Povolotsky, Dec 25 2008, is the case n = 0 of the more general result (obtained by the WZ method): for n >= 0, there holds
Pi = Sum_{j = 0.. n-1} 2^(j+1)/((2*j + 1)*binomial(2*j,j)) + 8*(n+1)!*Sum_{k >= 0} 1/((4*k + 1)*(4*k + 3)*...*(4*k + 2*n + 3)).
Letting n -> oo gives the rapidly converging series Pi = Sum_{j >= 0} 2^(j+1)/((2*j + 1)*binomial(2*j,j)) due to Euler.
More generally, it appears that for n >= 1, Pi = 1/(2*n-1)!!^2 * Sum_{j >= 0} (Product_{i = 0..2*n-1} j - i) * 2^(j+1)/((2*j + 1)*binomial(2*j,j)).
For any integer n, Pi = (-1)^n * 4 * Sum_{k >= 0} 1/(4*k + 1 + 2*n) - 1/(4*k + 3 - 2*n). (End)
Pi = Product_{k>=1} ((k^3*(k + 2)*(2*k + 1)^2)/((k + 1)^4*(2*k - 1)^2))^k. - Antonio Graciá Llorente, Jun 13 2024
Equals Integral_{x=0..2} sqrt(8 - x^2) dx - 2 (see Ambrisi and Rizzi). - Stefano Spezia, Jul 21 2024
Equals 3 + 4*Sum_{k>0} (-1)^(k+1)/(4*k*(1+k)*(1+2*k)) (see Wells at p. 53). - Stefano Spezia, Aug 31 2024
Equals 4*Integral_{x=0..1} sqrt(1 - x^2) dx = lim_{n->oo} (4/n^2)*Sum_{k=0..n} sqrt(n^2 - k^2) (see Finch). - Stefano Spezia, Oct 19 2024
Equals Beta(1/2,1/2) (see Shamos). - Stefano Spezia, Jun 03 2025
From Kritsada Moomuang, Jun 18 2025: (Start)
Equals 2 + Integral_{x=0..1} 1/(sqrt(x)*(1 + sqrt(1 - x))) dx.
Equals 2 + Integral_{x=0..1} log(1 + sqrt(1 - x))/sqrt(x) dx. (End)
Pi = 2*arccos(1/phi) + arccos(1/phi^3) = 4*arcsin(1/phi) + 2*arcsin(1/phi^3) where phi = (1+sqrt(5))/2. - Chittaranjan Pardeshi, Jul 02 2025
Pi = Sum_{n >= 0} zeta(2*n)*(2^(2*n - 1) - 1)/2^(4*n - 3). - Andrea Pinos, Jul 29 2025
Extensions
Additional comments from William Rex Marshall, Apr 20 2001
Comments