A053004 -id:A053004 - cp's OEIS frontend

A068465 Decimal expansion of Gamma(3/4).

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

Offset: 1

Benoit Cloitre, Mar 10 2002

			Gamma(3/4) = 1.225416702465177645129098303362890526851239248108070611...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:14 at page 414.

G. C. Greubel, Table of n, a(n) for n = 1..20000
Russell J. Matheson, GAMMA(3/4) computed to 14550 digits.
Simon Plouffe, GAMMA(3/4) to 256 places, see p. 65.
Index to sequences related to the Gamma function

Cf. A000796, A002193, A053004, A063448, A068466, A069998.

SetDefaultRealField(RealField(105)); Gamma(3/4); // G. C. Greubel, Mar 11 2018

evalf(GAMMA(3/4)) ; # R. J. Mathar, Jan 10 2013

RealDigits[Gamma[3/4], 10, 100][[1]] (* G. C. Greubel, Mar 11 2018 *)

default(realprecision, 100); gamma(3/4) \\ G. C. Greubel, Mar 11 2018

Gamma(3/4) * A068466 = sqrt(2)*Pi = A063448. - R. J. Mathar, Jun 18 2006
Equals Integral_{x>=0} x^(-1/4)*exp(-x) dx. - Vaclav Kotesovec, Nov 12 2020
Equals (Pi/2)^(1/4) * sqrt(AGM(1,sqrt(2))) = sqrt(A069998 * A053004). - Amiram Eldar, Jun 12 2021

A062539 Decimal expansion of the Lemniscate constant or Gauss's constant.

Original entry on oeis.org

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

Offset: 1

Jason Earls, Jun 25 2001

			2.622057554292119810464839589891119413682754951431623162816821703...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.3 and 6.2, pp. 99, 420.

Harry J. Smith, Table of n, a(n) for n = 1..5000
Lorenz Milla, World record computation of Lemniscate Constant (2,000,000,000,000 digits).
Simon Plouffe, Lemniscate or Gauss constant.
Simon Plouffe, Lemniscate constant or Gauss constant.
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Wikipedia, Lemniscate constant.
Index entries for transcendental numbers.

Cf. A062540, A064853, A085565.
Cf. A002193, A014549, A093341.
Equals A000796/A053004 (see PARI script).

SetDefaultRealField(RealField(100)); R:= RealField(); (1/2)*Sqrt(2*Pi(R)^3)/Gamma(3/4)^2; // G. C. Greubel, Oct 07 2018

evalf((1/2)*sqrt(2*Pi^3)/GAMMA(3/4)^2,120); # Muniru A Asiru, Oct 08 2018
evalf(1/2*GAMMA(1/4)*GAMMA(1/2)/GAMMA(3/4),120); # Martin Renner, Aug 16 2019
evalf(1/2*Beta(1/4,1/2),120); # Martin Renner, Aug 16 2019
evalf(2*int(1/sqrt(1-x^4),x=0..1),120); # Martin Renner, Aug 16 2019

RealDigits[Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2, 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

print(1/2*Pi^(3/2)/gamma(3/4)^2*2^(1/2))

allocatemem(932245000); default(realprecision, 5080); x=Pi^(3/2)*sqrt(2)/(2*gamma(3/4)^2); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062539.txt", n, " ", d)); \\ Harry J. Smith, Jun 20 2009

Pi/agm(1,sqrt(2)) \\ Charles R Greathouse IV, Feb 04 2015

intnum(x=0,Pi, 1/sqrt(1 + sin(x)^2)) \\ Charles R Greathouse IV, Feb 04 2025

Equals (1/2)*sqrt(2*Pi^3)/Gamma(3/4)^2.
A093341 multiplied by A002193. - R. J. Mathar, Aug 28 2013
From Martin Renner, Aug 16 2019: (Start)
Equals 2*Integral_{x=0..1} 1/sqrt(1-x^4) dx.
Equals 1/2*B(1/4,1/2) with Beta function B(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y). (End)
Equals Pi/AGM(1, sqrt(2)). - Jean-François Alcover, Feb 28 2021
Equals 2*hypergeom([1/2, 1/4], [5/4], 1). - Peter Bala, Mar 02 2022
Equals (1/2)*A064853 = 2*A085565. - Amiram Eldar, May 04 2022
Equals Pi*A014549. - Hugo Pfoertner, Jun 28 2024
Equals Integral_{x=0..Pi} 1/sqrt(1 + sin(x)^2) dx = EllipticK(-1) (see Finch at p. 420). - Stefano Spezia, Dec 15 2024
Equals Gamma(1/4)^2 / (sqrt(Pi)*2^(3/2)). - Vaclav Kotesovec, Apr 26 2025
Equals (161*6440^(1/4))/(2*Sum_{k>=0} N(k)/D(k)) with N(k) = Pochhammer(1/8,k) * Pochhammer(5/8,k) * (275+8640*k) and D(k) = (k!)^2*25921^k [Jorge Zuniga, 2023].

A014549 Decimal expansion of 1 / M(1,sqrt(2)) (Gauss's constant).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, N. J. A. Sloane

On May 30, 1799, Gauss discovered that this number is also equal to (2/Pi)*Integral_{t=0..1} 1/sqrt(1-t^4).

M(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).

			0.8346268416740731862814297327990468...

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, Sections 1.5.4 and 6.1, pp. 34, 420.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Markus Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures, The Ramanujan Journal volume 54, pages 1-27 (2021). See pp. 4 and 24.
Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 2.
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 10.
Eric Weisstein's World of Mathematics, Gauss's Constant.
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean.
Index entries for transcendental numbers.

Cf. A053002, A053003, A053004, A062539, A093341, A244644, A248557.

SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(Pi(R)/2)/Gamma(3/4)^2; // G. C. Greubel, Aug 17 2018

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

RealDigits[Gamma[1/4]^2/(2*Pi^(3/2)*Sqrt[2]), 10, 105][[1]] (* or: *)
RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2]], 10, 105][[1]] (* Jean-François Alcover, Dec 13 2011, updated Nov 11 2016, after Eric W. Weisstein *)
First[RealDigits[N[EllipticTheta[4, Exp[-Pi]]^2, 90]]] (* Stefano Spezia, Sep 29 2022 *)

default(realprecision, 20080); x=10*agm(1, sqrt(2))^-1; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b014549.txt", n, " ", d)); \\ Harry J. Smith, Apr 20 2009

1/agm(sqrt(2),1) \\ Charles R Greathouse IV, Feb 04 2015

sqrt(Pi/2)/gamma(3/4)^2 \\ Charles R Greathouse IV, Feb 04 2015

from mpmath import mp, agm, sqrt
mp.dps=105
print([int(z) for z in list(str(1/agm(sqrt(2)))[2:-1])]) # Indranil Ghosh, Jul 11 2017

Equals (lim_{k->oo} p(k))/(1+i) and (lim_{k->oo} q(k))/(1+i), where i is the imaginary unit, p(0) = 1, q(0) = i, p(k+1) = 2*p(k)*q(k)/(p(k)+q(k)) and q(k+1) = sqrt(p(k)*q(k)) for k >= 0. - A.H.M. Smeets, Jul 26 2018
Equals the infinite quotient product (3/4)*(6/5)*(7/8)*(10/9)*(11/12)*(14/13)*(15/16)*... . - James Maclachlan, Jul 28 2019
Equals (9/15)*hypergeom([1/2, 3/4], [9/4], 1). - Peter Bala, Mar 03 2022
Equals A062539 / Pi. - Amiram Eldar, May 04 2022
From Stefano Spezia, Sep 29 2022: (Start)
Equals theta4(exp(-Pi))^2.
Equals sqrt(2)*A093341/Pi. (End)
Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)^2/16^k. - Amiram Eldar, Jul 04 2023
From Gerry Martens, Jul 31 2023: (Start)
Equals 2*Gamma(5/4)/(sqrt(Pi)*Gamma(3/4)).
Equals hypergeom([1/4, -2/4], [1], 1). (End)
Equals A248557^2. - Hugo Pfoertner, Jun 28 2024

Extended to 105 terms by Jean-François Alcover, Dec 13 2011

a(104) corrected by Andrew Howroyd, Feb 23 2018

A076390 Decimal expansion of lemniscate constant B.

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

Offset: 0

Robert G. Wilson v, Oct 09 2002

Also decimal expansion of AGM(1,i)/(1+i).
See A085565 for the lemniscate constant A. - Peter Bala, Oct 25 2019
Also the ratio of height to diameter of a "Mylar balloon" (see Paulsen). - Jeremy Tan, May 05 2021

			0.599070117367796103719961246140161939113606331607825779131837476473202607...
AGM(1,i) = 0.59907011736779610371... + 0.59907011736779610371...*i

J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, 1998.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.

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, Vol. 18, No. 1 (1975), pp. 14-19; corrigendum, Vol. 18, No. 8 (1975), p. 462.
J. Todd, The lemniscate constants, in Pi: A Source Book, pp. 412-417.
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean.
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Wikipedia, Mylar balloon.
Wolfram Research, Arithmetic-Geometric Mean.

Cf. A053004, A076391, A076392, A085565, A243340.

RealDigits[ Chop[ N[ ArithmeticGeometricMean[1, I]/(1 + I), 111]]] [[1]]
RealDigits[N[Pi/(4 EllipticK[-1]), 106]][[1]] (* Jean-François Alcover, Jun 02 2019 *)

real(agm(1,I)/(1+I)) \\ Charles R Greathouse IV, Mar 03 2016

(2*Pi)^(-1/2)*gamma(3/4)^2 \\ Michel Marcus, Nov 10 2017

Pi/4/ellK(I) \\ Charles R Greathouse IV, Feb 04 2025

Equals (2*Pi)^(-1/2)*GAMMA(3/4)^2.
Equals ee/sqrt(2)-1/2*sqrt(2*ee^2-Pi) where ee = EllipticE(1/2), or also Product_{m>=1} ((2*m)/(2*m-1))^(-1)^m. - Jean-François Alcover, Sep 02 2014, after Steven Finch.
Equals sqrt(2) * Pi^(3/2) / GAMMA(1/4)^2. - Vaclav Kotesovec, Oct 03 2019
From Peter Bala, Oct 25 2019: (Start)
Equals 1 - 1/3 - 1/(3*7) - (1*3)/(3*7*11) - (1*3*5)/(3*7*11*15) - ... = hypergeom([-1/2,1],[3/4],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 - 1)*r(n-1) with r(0) = 1. Then the constant equals Sum_{n >= 0} r(n) = 1 - 1/3 - 1/21 - 1/77 - 1/231 - 1/627 - 3/4807 - 1/3933 - 13/121923 - 13/284487 - 17/853461 - .... The partial sum of the series to 100 terms gives the constant correct to 32 decimal places.
Equals (1/3) + (1*3)/(3*7) + (1*3*5)/(3*7*11) + ... = (1/3) * hypergeom ([3/2,1],[7/4],1/2). (End)
Equals (1/2) * A053004. - Amiram Eldar, Aug 26 2020
Equals (2/3) * 1/A243340. - Peter Bala, Mar 25 2024
Equals Product_{n>=1} exp(((-1)^n*beta(n))/n), where beta(n) is the Dirichlet beta function. - Antonio Graciá Llorente, Oct 16 2024
Equals Integral_{x=0..1} x^2/sqrt(1 - x^4) dx = sqrt(Pi)*Gamma(7/4)/(3*Gamma(5/4)) (see Finch). - Stefano Spezia, Dec 15 2024

Edited by N. J. A. Sloane, Nov 01 2008 at the suggestion of R. J. Mathar

A096427 Decimal expansion of 1/(sqrt(2)*G), where G is Gauss's constant A014549.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2004

Also, decimal expansion of Product_{n>=1} (1-1/(4n-1)^2). - Bruno Berselli, Apr 02 2013

			0.8472130847939790866064991234821916364814459103269... = agm(1, sqrt(1/2))

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equation 1:7:6 at page 13.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Peter Bala, Notes on the constants A096427 and A224268 .
Eric Weisstein's World of Mathematics, Gauss's Constant.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Ubiquitous Constant.
Index entries for transcendental numbers.

Cf. A014549, A062539, A224268, A091670 (1/C^2), A175574 (1/C), A293238 (C^2), A053004 (sqrt(2)*C), A327995.

SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(3/4)^2/(Sqrt(2)*Sqrt(Pi(R)/2)); // G. C. Greubel, Aug 17 2018

RealDigits[ArithmeticGeometricMean[1, Sqrt[2]]/Sqrt[2], 10, 110][[1]] (* Bruno Berselli, Apr 02 2013 *)
(* From the comment: *) RealDigits[N[Product[1 - 1/(4 n - 1)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

{ default(realprecision, 20080); x=agm(1, sqrt(1/2)); d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b096427.txt", n, " ", d)); } \\ Harry J. Smith, Apr 15 2009

agm(1, sqrt(1/2)) \\ Michel Marcus, Jun 09 2019

Also equals agm(1,1/sqrt(2)) since agm(1,1/b) = (1/b)*agm(1,b). - Gerald McGarvey, Sep 22 2008
From Peter Bala, Feb 26 2019: (Start)
C = Gamma(3/4)^2/sqrt(Pi).
C = 1/( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.
C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} (-1)^n*exp(-Pi*n^2 ) )^2.
Conjecturally, C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} exp(-Pi*(n+1/2)^2 ) )^2.
C = ((-1)^m*4^m/binomial(2*m,m)) * Product_{n >= 0} ( 1 - (4*m + 1)^2/(4*n + 3)^2 ), for m = 0,1,2,....
C = 1 - Integral_{x = 0..1} (sqrt(1 + x^4) - 1)/x^2 dx.
C = 1 - Sum_{n >= 1} binomial(1/2,n)/(4*n - 1) = 1 - Sum_{n >= 0} (-1)^n/(4*n + 3)*Catalan(n)/2^(2*n + 1).
Continued fraction: 1 - 1/(3 + 6/(1 + 12/(3 + ... + (4*n - 1)*(4*n - 2)/(1 + 4*n*(4*n - 1)/(3 + ... ))))). (End)
From Peter Bala, Mar 02 2022 : (Start)
C = (2/3)*hypergeom([1/4, 3/4], [7/4], 1)
C = hypergeom([-1/4, 1/4], [3/4], 1).
C = hypergeom([-1/2, -1/4], [3/4], -1). Cf. A053004.
C = (16/21)*hypergeom([-1/4, -3/4], [7/4], 1). (End)
Equals Pi/(sqrt(2)*A062539). - Amiram Eldar, May 04 2022
C = Integral_{x = 0..Pi/2} sqrt(sin(x)*cos(x)) dx. - Adam Hugill, Nov 27 2022
Equals 1/A175574 = sqrt(A293238) = A327995^2. - Hugo Pfoertner, Dec 26 2024

A335930 Decimal expansion of the arclength on y = sin(x) from (0,0) to (Pi,0).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 01 2020

Also the arclength between consecutive points of intersection of y = sin(x) and y = cos(x).

			arclength = 3.8201977890277120179047620821714432919099676146...

Cf. A335928, A335929, A335931, A335932.

Cf. A000796, A010466, A053004, A062539, A105419, A256667, A257407.

r = NIntegrate[Sqrt[1 + Cos[t]^2], {t, 0, Pi}, WorkingPrecision -> 200]
RealDigits[r][[1]]
First[RealDigits[Sqrt[8]*EllipticE[1/2], 10, 100]] (* Paolo Xausa, Nov 14 2024 *)

From Paolo Xausa, Nov 14 2024: (Start)
Equals Pi/A062539 + A062539 = A053004 + A062539.
Equals A010466*A257407. (End)
Equals A105419/2 = 2*A256667. - Hugo Pfoertner, Nov 14 2024

A053003 Continued fraction for M(1,sqrt(2)).

Original entry on oeis.org

1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1, 2, 2, 6, 9, 1, 1, 1, 3, 1, 2, 6, 1, 5, 1, 1, 2, 1, 13, 2, 2, 5, 1, 2, 2, 1, 5, 1, 3, 1, 3, 1, 2, 2, 2, 2, 8, 3, 1, 2, 2, 1, 10, 2, 2, 2, 3, 3, 1, 7, 1, 8, 3, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 2, 17, 1, 4, 31, 2, 2, 5, 30, 1, 8, 2, 1

Offset: 0

N. J. A. Sloane, Feb 21 2000

M(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...

J. M. Borwein and P. B. Borwein, Pi and the AGM, page 5.
J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

Harry J. Smith, Table of n, a(n) for n = 0..19999
Eric Weisstein's World of Mathematics, Gauss's Constant.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A014549, A053002 without the leading term, A053004 (decimal expansion).

ContinuedFraction[ArithmeticGeometricMean[1,Sqrt[2]],100] (* Harvey P. Dale, Feb 26 2012 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(agm(1, sqrt(2))); for (n=1, 20000, write("b053003.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

More terms from James Sellers, Feb 22 2000

Offset changed by Andrew Howroyd, Aug 03 2024

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

A309893 Decimal expansion of AGM(1, sqrt(3)/2).

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Aug 21 2019

Related to the pendulum acceleration relation at 60 degrees. In general, the period T of a mathematical pendulum with a maximum deflection angle theta is 2*Pi*sqrt(L/g)/AGM(1, cos(theta/2)), where L is the length of the pendulum, g is the gravitational acceleration, and 0 < theta <= 90 degrees. For theta = 60 degrees, the period is T = 2*Pi*sqrt(L/g)/AGM(1, sqrt(3)/2). - Jianing Song, Nov 21 2022

			0.931808391622448271177844...

Wikipedia, Pendulum (mechanics)

Cf. A310000 (AGM(1, cos(Pi/5))), A096427 (AGM(1, sqrt(2)/2)), A053004, A014549, A068521.

RealDigits[ArithmeticGeometricMean[1, Sqrt[3]/2], 10, 100][[1]] (* Amiram Eldar, Aug 21 2019 *)

agm(1, sqrt(3)/2) \\ Michel Marcus, Aug 22 2019

import decimal
prec = int(input('Precision: '))
decimal.getcontext().prec = prec
D = decimal.Decimal
def agm(a, b):
    for x in range(prec):
        a, b = (a + b) / 2,(a * b).sqrt()
    return a
print(agm(1, D(3).sqrt()/2))

RealField(300)(1.0).agm(sqrt(3)/2) # Peter Luschny, Aug 22 2019

AGM(1, sin(Pi/3)).

A310000 Decimal expansion of AGM(1, phi/2), where phi is the golden ratio (A001622).

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Aug 26 2019

Related to the pendulum acceleration relation at 72 degrees. 2*Pi*sqrt(l/g)/AGM(1, phi/2) gives the period T of a mathematical pendulum with a maximum deflection angle of 72 degrees from the downward vertical. The length of the pendulum is l and g is the gravitational acceleration.

			0.9019793381143431233972715365...

Cf. A001622, A309893, A053004, A014549, A068521.

RealDigits[ArithmeticGeometricMean[1, GoldenRatio/2], 10, 100][[1]] (* Amiram Eldar, Aug 26 2019 *)

agm(1, cos(Pi/5)) \\ Michel Marcus, Apr 05 2020

import decimal
iters = int(input('Precision: '))
decimal.getcontext().prec = iters
D = decimal.Decimal
def agm(a, b):
    for x in range(iters):
        a, b = (a + b) / 2,(a * b).sqrt()
    return a
print(agm(1, (D(5).sqrt()+1)/4))

Equals AGM(1, cos(Pi/5)).

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A068465 Decimal expansion of Gamma(3/4).

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A062539 Decimal expansion of the Lemniscate constant or Gauss's constant.

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A014549 Decimal expansion of 1 / M(1,sqrt(2)) (Gauss's constant).

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A076390 Decimal expansion of lemniscate constant B.

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A096427 Decimal expansion of 1/(sqrt(2)*G), where G is Gauss's constant A014549.

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