A076390 -id:A076390 - cp's OEIS frontend

A076391 Continued fraction for agm(1,i)/(1+i) (A076390).

0, 1, 1, 2, 42, 1, 1, 1, 1, 1, 1, 6, 1, 4, 3, 1, 1, 32, 1, 1, 16, 18, 2, 1, 10, 1, 3, 2, 2, 1, 2, 1, 1, 4, 3, 8, 2, 1, 2, 1, 12, 6, 1, 6, 4, 1, 11, 2, 2, 12, 1, 1, 2, 1, 1, 5, 1, 4, 1, 16, 1, 1, 5, 1, 2, 5, 4, 1, 4, 1, 1, 1, 1, 2, 3, 1, 17, 1, 1, 4, 4, 3, 2, 1, 2, 5, 8, 1, 9, 15, 1, 2, 2, 1, 2, 61, 1, 3

Offset: 1

Robert G. Wilson v, Oct 09 2002

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000
Wolfram Research, Arithmetic-Geometric Mean
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean

Cf. A076390 & A076392.

ContinuedFraction[ Chop[ N[ ArithmeticGeometricMean[1, I]/(1 + I), 150]]]

A085566 Duplicate of A076390.

Original entry on oeis.org

5, 9, 9, 0, 7, 0, 1, 1, 7, 3, 6, 7, 7, 9, 6, 1, 0, 3, 7, 1, 9, 9, 6, 1, 2, 4, 6, 1, 4, 0, 1, 6, 1

Offset: 0

dead

A000796 Decimal expansion of Pi (or digits of Pi).

Original entry on oeis.org

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

N. J. A. Sloane

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} (-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)
Sometimes referred to as Archimedes's constant, because the Greek mathematician computed lower and upper bounds of Pi by drawing regular polygons inside and outside a circle. In Germany it was called the Ludolphian number until the early 20th century after the Dutch mathematician Ludolph van Ceulen (1540-1610), who calculated up to 35 digits of Pi in the late 16th century. - Martin Renner, Sep 07 2016
As of the beginning of 2019 more than 22 trillion decimal digits of Pi are known. See the Wikipedia article "Chronology of computation of Pi". - Harvey P. Dale, Jan 23 2019
On March 14, 2019, Emma Haruka Iwao announced the calculation of 31.4 trillion digits of Pi using Google Cloud's infrastructure. - David Radcliffe, Apr 10 2019
Also volume of three quarters of a sphere of radius 1. - Omar E. Pol, Aug 16 2019
On August 5, 2021, researchers from the University of Applied Sciences of the Grisons in Switzerland announced they had calculated 62.8 trillion digits. Guinness World Records has not verified this yet. - Alonso del Arte, Aug 23 2021
The Hermite-Lindemann (1882) theorem states, that if z is a nonzero algebraic number, then e^z is a transcendent number. The transcendence of Pi then results from Euler's relation: e^(i*Pi) = -1. - Peter Luschny, Jul 21 2023
The 10000 words of the book "Not A Wake" by Michael Keith, written in Pilish, match in length the first 10000 digits of Pi. - Paolo Xausa, Aug 07 2025

			3.1415926535897932384626433832795028841971693993751058209749445923078164062\
862089986280348253421170679821480865132823066470938446095505822317253594081\
284811174502841027019385211055596446229489549303819...

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.

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

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).
Cf. A001622, A048940, A248682.

-- 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

-- 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

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 */

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // Bruno Berselli, Mar 12 2013

Digits := 110: Pi*10^104:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Oct 29 2019

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+ *)

{ 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

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

first(n)= default(realprecision, n+10); digits(floor(Pi*10^(n-1))) \\ David A. Corneth, Jun 21 2022

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

from sympy import pi, N; print(N(pi, 1000)) # David Radcliffe, Apr 10 2019

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

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

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

Additional comments from William Rex Marshall, Apr 20 2001

A085565 Decimal expansion of lemniscate constant A.

Original entry on oeis.org

1, 3, 1, 1, 0, 2, 8, 7, 7, 7, 1, 4, 6, 0, 5, 9, 9, 0, 5, 2, 3, 2, 4, 1, 9, 7, 9, 4, 9, 4, 5, 5, 5, 9, 7, 0, 6, 8, 4, 1, 3, 7, 7, 4, 7, 5, 7, 1, 5, 8, 1, 1, 5, 8, 1, 4, 0, 8, 4, 1, 0, 8, 5, 1, 9, 0, 0, 3, 9, 5, 2, 9, 3, 5, 3, 5, 2, 0, 7, 1, 2, 5, 1, 1, 5, 1, 4, 7, 7, 6, 6, 4, 8, 0, 7, 1, 4, 5, 4

Offset: 1

N. J. A. Sloane, Jul 06 2003

This number is transcendental by a result of Schneider on elliptic integrals. - Benoit Cloitre, Jan 08 2006
The two lemniscate constants are A = Integral_{x = 0..1} 1/sqrt(1 - x^4) dx and B = Integral_{x = 0..1} x^2/sqrt(1 - x^4) dx. See A076390. - Peter Bala, Oct 25 2019
Also the ratio of generating curve length to diameter of a "Mylar balloon" (see Paulsen). - Jeremy Tan, May 05 2021

			1.3110287771460599052324197949455597068413774757158115814084108519...

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
Th. Schneider, Transzendenzuntersuchungen periodischer Funktionen (1934).
Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale (1937).

G. C. Greubel, Table of n, a(n) for n = 1..5000
John Maxwell Campbell, WZ proofs for lemniscate-like constant evaluations, Integers 21 (2021), Article A107, 15.
S. Khrushchev, Orthogonal polynomials and continued fractions from Euler’s point of view, Encyclopedia of Mathematics and its Applications 122.
Rensley Meulens, A note on N-soliton solutions for the viscid incompressible Navier-Stokes differential equation, AIP Advances (2022) Vol. 12, 015308.
W. H. Paulsen, What Is the Shape of a Mylar Balloon?, Amer. Math. Monthly 101 (10), (Dec. 1994), pp. 953-958.
J. Todd, The lemniscate constants, Comm. ACM, 18 (1975), 14-19; 18 (1975), 462.
J. Todd, The lemniscate constants, in Pi: A Source Book, pp. 412-417.
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Wikipedia, Mylar balloon.
Index entries for transcendental numbers.

Cf. A076390, A225119, A243340.

C := ComplexField(); [Gamma(1/4)^2/(4*Sqrt(2*Pi(C)))]; // G. C. Greubel, Nov 05 2017

Mathematica

RealDigits[ Gamma[1/4]^2/(4*Sqrt[2*Pi]), 10, 99][[1]] (* or *) RealDigits[ EllipticK[-1], 10, 99][[1]] (* Jean-François Alcover, Mar 07 2013, updated Jul 30 2016 *)

PARI

gamma(1/4)^2/4/sqrt(2*Pi)

PARI

K(x)=Pi/2/agm(1,sqrt(1-x)) K(-1) \\ Charles R Greathouse IV, Aug 02 2018

PARI

ellK(I) \\ Charles R Greathouse IV, Feb 04 2025

Formula

Equals (1/4)*(2*Pi)^(-1/2)*GAMMA(1/4)^2.

Equals Integral_{x>=1}dx/sqrt(4x^3-4x). - Benoit Cloitre, Jan 08 2006

Equals Product_(k>=0, [(4k+3)(4k+4)] / [(4k+5)(4k+2)] ) (Gauss). - Ralf Stephan, Mar 04 2008 [corrected by Vaclav Kotesovec, May 01 2020]

Equals Pi/sqrt(8)/agm(1,sqrt(1/2)).

Equals Pi/sqrt(8)*hypergeom([1/2,1/2],[1],1/2).

Product_{m>=1} ((2*m)/(2*m+1))^(-1)^m. - Jean-François Alcover, Sep 02 2014, after Steven Finch

From Peter Bala, Mar 09 2015: (Start)

Equals Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.

Continued fraction representations: 2/(1 + 1*3/(2 + 5*7/(2 + 9*11/(2 + ... )))) due to Euler - see Khrushchev, p. 179.

Also equals 1 + 1/(2 + 2*3/(2 + 4*5/(2 + 6*7/(2 + ... )))). (End)

From Peter Bala, Oct 25 2019: (Start)

Equals 1 + 1/5 + (1*3)/(5*9) + (1*3*5)/(5*9*13) + ... = hypergeom([1/2,1],[5/4],1/2) by Gauss's second summation theorem.

Equivalently, define a sequence of rational numbers r(n) recursively by r(n) = (2*n - 3)/(4*n - 3)*r(n-1) with r(1) = 1. Then the constant equals Sum_{n >= 1} r(n) = 1 + 1/5 + 1/15 + 1/39 + 7/663 + 1/221 + 11/5525 + 11/12325 + 1/2465 + .... The partial sum of the series to 100 terms gives 32 correct decimal digits for the constant.

Equals (1*3)/(1*5) + (1*3*5)/(1*5*9) + (1*3*5*7)/(1*5*9*13) + ... = (3/5) * hypergeom([5/2,1],[9/4],1/2). (End)

Equals (3/2)*A225119. - Peter Bala, Oct 27 2019

Equals Integral_{x=0..Pi/2} 1/sqrt(1 + cos(x)^2) dx = Integral_{x=0..Pi/2} 1/sqrt(1 + sin(x)^2) dx. - Amiram Eldar, Aug 09 2020

From Peter Bala, Mar 24 2024: (Start)

An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 1 for k >= 0.

For example, taking k = 0 and k = 1 yields

A = 2/(1 + (1*3)/(2 + (5*7)/(2 + (9*11)/(2 + (13*15)/(2 + ... + (4*n + 1)*(4*n + 3)/(2 + ... )))))) and

A = (1/4)*(5 + (1*3)/(10 + (5*7)/(10 + (9*11)/(10 + (13*15)/(10 + ... + (4*n + 1)*(4*n + 3)/(10 + ... )))))). (End)

A053004 Decimal expansion of AGM(1,sqrt(2)).

Original entry on oeis.org

1, 1, 9, 8, 1, 4, 0, 2, 3, 4, 7, 3, 5, 5, 9, 2, 2, 0, 7, 4, 3, 9, 9, 2, 2, 4, 9, 2, 2, 8, 0, 3, 2, 3, 8, 7, 8, 2, 2, 7, 2, 1, 2, 6, 6, 3, 2, 1, 5, 6, 5, 1, 5, 5, 8, 2, 6, 3, 6, 7, 4, 9, 5, 2, 9, 4, 6, 4, 0, 5, 2, 1, 4, 1, 4, 3, 9, 1, 5, 6, 7, 0, 8, 3, 5, 8, 8, 5, 5, 5, 6, 4, 8, 9, 7, 9, 3, 3, 8, 9, 3, 7, 5, 9, 0

Offset: 1

N. J. A. Sloane, Feb 21 2000

AGM(a,b) is the limit of the arithmetic-geometric mean iteration applied repeatedly starting with a and b: a_0=a, b_0=b, a_{n+1}=(a_n+b_n)/2, b_{n+1}=sqrt(a_n*b_n).

			1.19814023473559220743992249228...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, page 5.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 420.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Gauss's Constant.
Index entries for transcendental numbers.

Cf. A014549, A053002, A053003, A076390, A085565, A096427.

evalf(GaussAGM(1, sqrt(2)), 144);  # Alois P. Heinz, Jul 05 2023

RealDigits[ N[ ArithmeticGeometricMean[1, Sqrt[2]], 105]][[1]] (* Jean-François Alcover, Jan 30 2012 *)
RealDigits[N[(1+Sqrt[2])Pi/(4EllipticK[17-12Sqrt[2]]), 105]][[1]] (* Jean-François Alcover, Jun 02 2019 *)

default(realprecision, 20080); x=agm(1, sqrt(2)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b053004.txt", n, " ", d)) \\ Harry J. Smith, Apr 20 2009

2*real(agm(1, I)/(1+I)) \\ Michel Marcus, Jul 26 2018

from mpmath import mp, agm, sqrt
mp.dps=106
print([int(z) for z in list(str(agm(1, sqrt(2))).replace('.', '')[:-1])]) # Indranil Ghosh, Jul 11 2017

Equals Pi/(2*A085565). - Nathaniel Johnston, May 26 2011
Equals Integral_{x=0..Pi/2} sqrt(sin(x)) or Integral_{x=0..1} sqrt(x/(1-x^2)). - Jean-François Alcover, Apr 29 2013 [cf. Boros & Moll p. 195]
Equals Product_{n>=1} (1+1/A033566(n)) and also 2*AGM(1, i)/(1+i) where i is the imaginary unit. - Dimitris Valianatos, Oct 03 2016
Conjecturally equals 1/( Sum_{n = -infinity..infinity} exp(-Pi*(n+1/2)^2 ) )^2. Cf. A096427. - Peter Bala, Jun 10 2019
From Amiram Eldar, Aug 26 2020: (Start)
Equals 2 * A076390.
Equals Integral_{x=0..Pi} sin(x)^2/sqrt(1 + sin(x)^2) dx. (End)
Equals sqrt(2/Pi)*Gamma(3/4)^2 = Integral_{x = 0..1} 1/(1 - x^2)^(1/4) dx = Pi/Integral_{x = 0..1} 1/(1 - x^2)^(3/4) dx. - Peter Bala, Jan 05 2022
From Peter Bala, Mar 02 2022: (Start)
Equals 2*Integral_{x = 0..1} x^2/sqrt(1 - x^4) dx.
Equals 1 - Integral_{x = 0..1} (sqrt(1 - x^4) - 1)/x^2 dx.
Equals hypergeom([-1/2, -1/4], [3/4], 1) = 1 + Sum_{n >= 0} 1/(4*n + 3)*Catalan(n)*(1/2^(2*n+1)). Cf. A096427. (End)

More terms from James Sellers, Feb 22 2000

A243340 Decimal expansion of 4L/(3Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.

Original entry on oeis.org

1, 1, 1, 2, 8, 3, 5, 7, 8, 8, 8, 9, 8, 7, 6, 4, 2, 4, 8, 3, 7, 5, 2, 3, 9, 6, 4, 3, 7, 3, 2, 0, 6, 2, 4, 1, 1, 9, 9, 1, 9, 9, 0, 6, 8, 4, 6, 5, 3, 7, 9, 6, 0, 0, 3, 2, 6, 6, 4, 3, 6, 4, 9, 3, 4, 7, 1, 5, 7, 5, 9, 9, 0, 2, 7, 9, 3, 6, 8, 5, 4, 9, 1, 5, 9, 5, 8, 8, 2, 1, 3, 8, 0, 1, 7, 0, 0, 4, 3, 2, 1, 7, 2, 0, 9

Offset: 1

Jean-François Alcover, Jun 03 2014

			1.11283578889876424837523964373206241199199...

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 102.

Eric Weisstein's World of Mathematics, Lemniscate Constant.

Cf. A062539 (L), A076390, A085565, A225119 (L/3).

L = Pi^(3/2)/(Sqrt[2]*Gamma[3/4]^2); RealDigits[4*L/(3*Pi), 10, 103] // First

2*sqrt(2*Pi)/(3*gamma(3/4)^2) \\ Stefano Spezia, Nov 27 2024

Equals 2*sqrt(2*Pi)/(3*Gamma(3/4)^2).
From  Peter Bala, Mar 24 2024: (Start)
An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 3 for k >= 0.
For example, taking k = 0 and k = 1 yields
4*L/(3*Pi) = 1 + 1/(6 + (5*7)/(6 + (9*11)/(6 + (13*15)/(6 + ... + (4*n + 1)*(4*n + 3)/(6 + ... ))))) and
4*L/(3*Pi) = 8/(7 + (1*3)/(14 + (5*7)/(14 + (9*11)/(14 + (13*15)/(14 + ... + (4*n + 1)*(4*n + 3)/(14 + ... )))))).
Equals (2/3) * 1/A076390. (End)

A076392 Increasing partial quotients of the continued fraction for agm(1,i)/(1+i).

Original entry on oeis.org

0, 1, 2, 42, 61, 88, 238, 254, 288, 347, 575, 4034, 9853, 21798, 49736, 108435, 109003, 181562, 1035352, 1955976, 6950275, 30712753, 41463747, 45117343, 112401242, 116579541

Offset: 1

Robert G. Wilson v, Oct 09 2002

			A076391(1) = 0
A076391(2) = 1
A076391(4) = 2
A076391(5) = 42
A076391(96) = 61
A076391(121) = 88
A076391(310) = 238
A076391(461) = 254
A076391(540) = 288
A076391(627) = 347
A076391(699) = 575
A076391(1136) = 4034
A076391(2986) = 9853
A076391(4172) = 21798
A076391(16727) = 49736
A076391(39201) = 108435
A076391(110180) = 109003
A076391(130606) = 181562
A076391(506314) = 1035352
A076391(512390) = 1955976
A076391(1248836) = 6950275
A076391(1990391) = 30712753
A076391(2528055) = 41463747
A076391(4853400) = 45117343
A076391(7427594) = 112401242
A076391(96166990) = 116579541

Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean
Wolfram Research, Arithmetic-Geometric Mean

Cf. A076390 & A076391.

a = ContinuedFraction[ Chop[ N[ ArithmeticGeometricMean[1, I]/(1 + I), 10^4]]]; b = 0; Do[ If[ a[[n]] > b, Print[a[[n]]]; b = a[[n]]], {n, 1, 10^4}]

a(21)-a(26) from Vaclav Kotesovec, Oct 03 2019

A105419 Decimal expansion of the arc length of the sine or cosine curve for one full period.

Original entry on oeis.org

7, 6, 4, 0, 3, 9, 5, 5, 7, 8, 0, 5, 5, 4, 2, 4, 0, 3, 5, 8, 0, 9, 5, 2, 4, 1, 6, 4, 3, 4, 2, 8, 8, 6, 5, 8, 3, 8, 1, 9, 9, 3, 5, 2, 2, 9, 2, 9, 4, 5, 4, 9, 4, 4, 2, 1, 6, 0, 9, 9, 3, 3, 1, 3, 4, 9, 4, 3, 9, 1, 6, 0, 2, 4, 2, 8, 6, 5, 9, 8, 4, 2, 1, 3, 2, 3, 6, 2, 1, 7, 8, 9, 0, 2, 4, 4, 4, 9, 6, 5, 6, 4, 4, 0, 8

Offset: 1

Robert G. Wilson v, Apr 06 2005

			I=7.640395578055424035809524164342886583819935229294549442160993313...

Howard Anton, Irl C. Bivens, Stephen L. Davis, Calculus, Early Transcendentals, 7th Edition, John Wiley & Sons, Inc., NY, Section 7.4 Length of a Plane Curve, page 489.

evalf(4*sqrt(2)*EllipticE(1/sqrt(2)), 120); # Vaclav Kotesovec, Apr 22 2015

RealDigits[ NIntegrate[ Sqrt[1 + Cos[x]^2], {x, 0, 2Pi}, MaxRecursion -> 12, WorkingPrecision -> 128], 10, 111][[1]]
RealDigits[ N[ 4*Sqrt[2]*EllipticE[1/2], 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Equals Integral_{x=0..2*Pi} sqrt(1+cos(x)^2) dx.

Also equals 4*B+Pi/B where B is the lemniscate constant A076390, or sqrt(2/Pi)*(2*gamma(3/4)^4 + Pi^2)/gamma(3/4)^2. - Jean-François Alcover, Apr 17 2013

A257407 Decimal expansion of E(1/sqrt(2)) = 1.35064..., where E is the complete elliptic integral.

Original entry on oeis.org

1, 3, 5, 0, 6, 4, 3, 8, 8, 1, 0, 4, 7, 6, 7, 5, 5, 0, 2, 5, 2, 0, 1, 7, 4, 7, 3, 5, 3, 3, 8, 7, 2, 5, 8, 4, 1, 3, 4, 9, 5, 2, 2, 3, 6, 6, 9, 2, 4, 3, 5, 4, 5, 4, 5, 3, 2, 3, 2, 5, 3, 7, 0, 8, 8, 5, 7, 8, 7, 7, 8, 9, 0, 8, 3, 6, 1, 2, 7, 3, 6, 9, 0, 4, 0, 2, 3, 6, 0, 7, 7, 8, 2, 2, 4, 9, 1, 5, 6, 3, 6, 0, 9, 9, 4, 7

Offset: 1

Jean-François Alcover, Apr 22 2015

This constant is sometimes expressed as E(1/2), with a different convention of argument (Cf. Mathematica).

			1.3506438810476755025201747353387258413495223669243545453232537...

Jonathan Borwein, David H. Bailey, Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century, CRC Press (2008), p. 145.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Elliptic Integral of the Second Kind
Wikipedia, Complete elliptic integral of the second kind

Cf. A010466, A053004, A062539, A076390, A093341, A105419.

evalf(EllipticE(1/sqrt(2)),120); # Vaclav Kotesovec, Apr 22 2015

RealDigits[EllipticE[1/2], 10, 106] // First

Equals (4*B^2 + Pi)/(4*sqrt(2)*B), where B is the lemniscate constant A076390.
Equals Pi^(3/2)/Gamma(1/4)^2 + Gamma(1/4)^2/(8*Pi^(1/2)).
Equals (agm(1,sqrt(2))+Pi/agm(1,sqrt(2)))/sqrt(8) = (A053004+A062539)/A010466. - Gleb Koloskov, Jun 29 2021

A337354 a(n) is the numerator of Product_{i=0..n-1} (n-i)^((-1)^ceiling(i/2)).

Original entry on oeis.org

1, 2, 3, 2, 5, 9, 7, 40, 45, 7, 308, 48, 975, 539, 88, 1664, 1105, 24255, 13376, 56576, 41769, 48279, 55936, 226304, 348075, 370139, 671232, 870400, 2082925, 4283037, 13872128, 80773120, 343682625, 4023459, 1553678336, 1900544, 14411758075, 59457783, 1471905792, 1406402560

Offset: 1

Devansh Singh, Aug 24 2020

a(n) is the numerator of (n/(n-1)) * ((n-3)/(n-2)) * ((n-4)/(n-5)) ...

			a(n)/A337355(n) equals 1, 2, 3/2, 2/3, 5/6, 9/5, 7/5, 40/63, 45/56, 7/4 ...
a(4) = numerator of (4*1)/(3*2) = numerator of 2/3 = 2.
a(5) = numerator of (5*2)/(4*3) = numerator of 5/6 = 5.
                      12  *   9*8  *  5*4  *  1
a(12) = numerator of --------------------------- = 48.
                        11*10  *  7*6  *  3*2

Cf. A337355 (denominators).

Cf. A076390, A085565, A007662.

a(n) = {numerator(prod(i=0, n-1, (n-i)^(-1)^((i+1)\2)))} \\ Andrew Howroyd, Aug 24 2020

a(n) = numerator of (n*A337355(n-2))/(a(n-2)*(n-1)) for n>=3.

Conjecture: a(4*n)/A337355(4*n) ~ 0.5990701173677... (=A076390). - Andrew Howroyd, Aug 25 2020

Terms a(31) and beyond from Andrew Howroyd, Aug 25 2020

cp's OEIS Frontend

A076391 Continued fraction for agm(1,i)/(1+i) (A076390).

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A085566 Duplicate of A076390.

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A000796 Decimal expansion of Pi (or digits of Pi).

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A085565 Decimal expansion of lemniscate constant A.

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A053004 Decimal expansion of AGM(1,sqrt(2)).

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A243340 Decimal expansion of 4*L/(3*Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.

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A243340 Decimal expansion of 4L/(3Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.