A093341 -id:A093341 - cp's OEIS frontend

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

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

A175576 Decimal expansion Pi^(3/2)/Gamma(3/4)^2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jul 15 2010

Entry 34 e of chapter 11 of Ramanujan's second notebook.
In addition, Pi^(3/2) / Gamma(3/4)^2 is the area of the unit "squircle" as defined in MathWorld. (Note that 8*Gamma(5/4)^2 / sqrt(Pi) is the same constant.) - Jean-François Alcover, Feb 24 2011
Real period of the elliptic curve y^2 = x*(x - 1)*(x - 1/2). See Rouse. - Peter Bala, Dec 06 2024

			3.708149354602743836867700694390520092435197647...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Bruce C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. Lond. Math. Soc. vol 15 no 4 (1983) 273-320.
Jeremy Rouse, Hypergeometric functions and elliptic curves, Ramanujan Journal, Vol. 12 (2006), pp. 197-205.
Eric Weisstein's World of Mathematics, Squircle

Cf. A062539, A068465, A175476, A175575, A290570.

Maple
```
Pi^(3/2)/GAMMA(3/4)^2 ; evalf(%) ;
```

RealDigits[Pi*EllipticTheta[3, 0, Exp[-Pi]]^2, 10, 50][[1]]
RealDigits[Pi^(3/2)/(Gamma[3/4])^2, 10, 50][[1]] (* G. C. Greubel, Feb 12 2017 *)

Pi^1.5/gamma(3/4)^2 \\ Charles R Greathouse IV, Jun 06 2016

Equals A175476 / A068465^2 = 1/A175575.
Equals Integral_{-oo, oo} 1/(1+2*x^2)^(3/4) or Integral_{-oo, oo} 1/sqrt(1+x^4). - Jean-François Alcover, Jun 04 2013
Equals sqrt(2)*L, where L is the lemniscate constant A062539. - Jean-François Alcover, Aug 11 2014
From Peter Bala, Mar 01 2022 : (Start)
Equals 3*Sum_{n >= 0} (1/(4*n+1) + 1/(4*n-3))*binomial(1/2,n). Cf. A290570.
Equals hypergeom([-1/2, 3/4, -3/4], [-1/4, 5/4], -1).
Equals 2*hypergeom([1/4, 3/4], [5/4], 1) = (16/5)*hypergeom([-1/4, -3/4], [5/4], 1). (End)
Equals 2 * A093341. - R. J. Mathar, Dec 08 2023
From Peter Bala, Dec 06 2024: (Start)
Equals Pi*hypergeom([1/2, 1/2], [1], 1/2).
Equals 2*Integral_{x = 0..Pi/2} 1/sqrt(1 - (1/2)*sin^2(x)) dx. See Rouse. (End)

A224268 Decimal expansion of Product_{n>=1} (1 - 1/(4n+1)^2).

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Apr 02 2013

			0.9270373386506859592169251735976300231087994117608834527929640225280...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Notes on the constants A096427 and A224268.

Cf. A064853, A085565, A093341, A327996.

Cf. product(1-1/(4n+r)^2, n>=1): A096427 (r=-1), A112628 (r=0), A179587-1 (r=2).

RealDigits[N[Product[1 - 1/(4 n + 1)^2, {n, 1, Infinity}], 90]][[1]] (* or, by the formula: *) RealDigits[Gamma[1/4]^2/(8 Sqrt[Pi]), 10, 90][[1]]

prodnumrat(1 - 1/(4*n+1)^2, 1) \\ Charles R Greathouse IV, Feb 07 2025

Equals Gamma(1/4)^2/(8*sqrt(Pi)) = L/(4*sqrt(2)), where L is the Lemniscate constant (A064853).
From Peter Bala, Feb 26 2019: (Start)
C = (Pi/4)*( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.
C = (-1)^m*2^(2*m+1)/Catalan(m) * Product_{n >= 1} ( 1 - (4*m + 3)^2/(4*n + 1)^2 ), for m = 0,1,2,....
C = Integral_{x = 0..1} 1/sqrt(1 + x^4) dx.
C = (1/sqrt(2))*Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.
C = (3/2)*Integral_{x = 0..1} sqrt(1 + x^4) dx - sqrt(2)/2.
C = (1/8)*Integral_{x = 0..1} 1/(x - x^2)^(3/4) dx.
C = Sum_{n >= 0} binomial(-1/2,n)/(4*n + 1) = Sum_{n >= 0} binomial(2*n,n)/4^n * 1/(4*n + 1).
C = (1/2)*Sum_{n >= 0} (-1)^n*binomial(-3/4,n)/(4*n + 1).
Continued fraction: 1 - 1/(5 + 20/(1 + 30/(3 + ... + (4*n)*(4*n + 1)/(1 + (4*n + 1)*(4*n + 2)/(3 + ... ))))).
C = A085565/sqrt(2). C = Pi/(4*A096427). (End)
Equals A093341/2 = A327996^2. - Hugo Pfoertner, Oct 31 2024

A064853 Decimal expansion of the Lemniscate constant.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 22 2001

			5.244115108584239620929679...

Harry J. Smith, Table of n, a(n) for n = 1..5000
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. 15.
Lorenz Milla, World record computation of Lemniscate Constant (2,000,000,000,000 digits)
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Eric Weisstein's World of Mathematics, Lemniscate.
Index entries for transcendental numbers.

Cf. A002193, A019727, A062539, A068466, A085565, A093341.

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

First@RealDigits[ N[ Gamma[ 1/4 ]^2/Sqrt[ 2 Pi ], 102 ] ]

{ allocatemem(932245000); default(realprecision, 5080); x=gamma(1/4)^2/sqrt(2*Pi); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b064853.txt", n, " ", d)); } \\ Harry J. Smith, Jun 20 2009

gamma(1/2)*gamma(1/4)/gamma(3/4) \\ Charles R Greathouse IV, Oct 29 2021

Equals Gamma(1/4)^2/sqrt(2*Pi). - G. C. Greubel, Oct 07 2018
Equals 2*A062539 = 4*A085565. - Amiram Eldar, May 04 2022
From Stefano Spezia, Sep 23 2022: (Start)
Equals 4*Integral_{x=0..Pi/2} 1/sqrt(2*(1 - (1/2)*sin(x)^2)) dx [Gauss, 1799] (see Faulhuber et al.).
Equals 2*sqrt(2)*A093341. (End)

A163973 Decimal expansion of Van der Pauw's constant = Pi/log(2).

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Aug 13 2009

Van der Pauw developed a method for measuring the sheet resistance of a four-terminal conducting sheet of arbitrary shape. Assuming the terminals to be point contacts at the periphery of the structure, he proved a general theorem that yields an analytical expression for the sheet resistance Rs. In the special case that the structure is invariant for a rotation of ninety degrees, the formula of Van der Pauw is Rs = (Pi/log(2))*(V/I).
A general theorem for the sheet resistance Rs of a Van der Pauw structure with finite contacts that is invariant for a rotation of ninety degrees was proved by Versnel. His theorem states that Rs = [K(k1)/K'(k1) - K(k2)/(2*K'(k2))]^(-1)*(V/I) with K(k) and K'(k) complete elliptic integrals with modulus k (Abramowitz and Stegun use parameter m = k^2).
Versnel found, with a little help from the author, expressions for Rs = C(d)*(V/I) for several Van der Pauw structures if d, the ratio of the sum of the lengths of the contacts and the length of the boundary of the sheet, tends to zero, see the formulas (first two terms are given). For point contacts, i.e., d = 0, Van der Pauw's constant appears.

			4.5323601418271938

G. C. Greubel, Table of n, a(n) for n = 1..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 17, pp. 589-626.
L.J. van der Pauw, A method of measuring specific resistivity and Hall effect of disc of arbitrary shape, Philips Research Reports, Vol. 13. no. 1, pp 1-9, February 1958.
W. Versnel, Analysis of symmetrical Van der Pauw structures with finite contacts, Solid State Electronics, Vol. 21, pp. 1261-1268, 1978.
W. Versnel, Analysis of the Greek cross, a Van der Pauw structure with finite contacts, Solid State Electronics, Vol. 22, pp. 911-914, 1979.
Eric. W. Weisstein, Elliptic Integral, from Wolfram MathWorld.
Wikipedia, Van der Pauw method

Cf. A000796 (Pi), A002162 (log(2)), A093341 (K), A131223 (2*Pi/log(2)), A259679 (log(2)/(4*Pi^2)).

RealDigits[N[Pi/Log[2], 103]][[1]] (* Mats Granvik, Apr 04 2012 *)

Pi/log(2) \\ Charles R Greathouse IV, Jan 30 2016

1) Circle with contacts in the middle of each side:
C(d) = Pi/log(2) + (Pi^3/(64*(log(2))^2))*d^2
2) Square with contacts in the middle of each side:
C(d) = Pi/log(2) + (Pi*K^2/(8*(log(2))^2))*d^2
3) Square with complementary contacts:
C(d) = Pi/log(2) + (Pi*K^4/(64*(log(2))^2))*d^4
with K = K(sqrt(2)/2) = 1.8540746773.
4) Greek cross with contacts at the cross ends:
C(d) = Pi/log(2) + 2*Pi/(log(2))^2*exp(Pi/2-Pi/d)
5) Greek cross with contacts between the cross ends:
C(d) = Pi/log(2) + ((Pi/(2^12*log(2)^2)*((-3/4)!/(-1/4)!)^8))*d^4

A249282 Decimal expansion of K(1/4), where K is the complete elliptic integral of the first kind.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 24 2014

			1.685750354812596042871203657799076989500800894141089...

Steven R. Finch, Gergonne-Schwarz Surface, April 12, 2013. [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.

Cf. A093341 (K(1/2)), A249283 (K(3/4)), A000796, A084895.

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

RealDigits[EllipticK[1/4], 10, 102] // First

ellK(1/2) \\ Charles R Greathouse IV, Feb 04 2025

From Paul D. Hanna, Mar 25 2024: (Start)
K(1/4) = Pi/2 * Sum_{n>=0} binomial(2*n,n)^2/16^n * (1/4)^n.
K(1/4) = Pi/2 * sqrt( Sum_{n>=0} binomial(2*n,n)^3/16^n * (m*(1-m))^n ), where m = 1/4. (End)
Equals Pi/agm(1, 3) = A000796 / A084895. - Amiram Eldar, Apr 28 2025

A225119 Decimal expansion of Integral_{x=0..Pi/2} sin(x)^(3/2) dx.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 29 2013

			0.87401918476403993682161319663037313789425165047720772093894...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant p. 102 and Section 6.1 Gauss' Lemniscate Constant p. 422.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A062539, A068466, A093341, A085565.

evalf((1/3)*sqrt(2)*EllipticK(1/sqrt(2)), 120); # Vaclav Kotesovec, Apr 22 2015

RealDigits[1/3*Sqrt[2]*EllipticK[1/2], 10, 100][[1]]

sqrt(Pi)*gamma(1/4)/(6*gamma(3/4)) \\ G. C. Greubel, Apr 01 2017

ellK(sqrt(1/2))*sqrt(2)/3 \\ Charles R Greathouse IV, Feb 04 2025

Equals 1/3 * sqrt(2) * ellipticK(1/2), (defined as in Mathematica).
Equals sqrt(2)/6 * Pi * hypergeom([1/2,1/2],[1],1/2).
Equals gamma(1/4)^2/(6*sqrt(2*Pi)).
Equals sqrt(Pi)*gamma(1/4)/(6*gamma(3/4)).
Equals Integral_{0..1} (1-x^2)^(1/4) dx.
Equals Integral_{0..1} sqrt(1-x^4) dx. - Charles R Greathouse IV, Aug 21 2017
Equals (2/3)*A085565. - Peter Bala, Oct 27 2019
Equals A062539/3. - Hugo Pfoertner, Dec 15 2024

A153396 G.f.: A(x) = F(xG(x)^3) where F(x) = G(x/F(x)^2) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)^2) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 5, 32, 228, 1726, 13587, 109923, 907499, 7609898, 64609346, 554108863, 4792190298, 41739160686, 365746143064, 3221723465187, 28509044813580, 253295607463902, 2258539046009268, 20203103111671575, 181242298665210280

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

			G.f.: A(x) = F(x*G(x)^3) = 1 + x + 5*x^2 + 32*x^3 + 228*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +...
G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 74*x^3 + 545*x^4 + 4228*x^5 +...
G(x)^3*A(x)^2 = 1 + 5*x + 32*x^2 + 228*x^3 + 1726*x^4 + 13587*x^5 +...

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A000108, A001764, A002293; A153395, A153397, A093341.

Join[{1},Table[Sum[Binomial[2k+1,k]/(2k+1) Binomial[4n-k,n-k]3 k/(4n-k), {k,0,n}],{n,20}]] (* Harvey P. Dale, Feb 09 2015 *)

{a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+1,k)/(2*k+1)*binomial(4*(n-k)+3*k,n-k)*3*k/(4*(n-k)+3*k)))}

a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(4n-k,n-k)*3k/(4n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^2 where G(x) is the g.f. of A002293.
G.f. satisfies: A(x/F(x)^2) = F(x*F(x)) where F(x) is the g.f. of A000108.
G.f. satisfies: A(x/H(x)) = F(x*H(x)^2) where H(x) = 1 + x*H(x)^3 is the g.f. of A001764 and F(x) is the g.f. of A000108.
G.f. satisfies: A(-x*A(x)^9) = 1/A(x). - Alexander Burstein, Apr 14 2020
Recurrence: 243*(n-1)*n*(n+1)*(3*n - 5)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(147456*n^6 - 1998336*n^5 + 11209920*n^4 - 33294250*n^3 + 55173779*n^2 - 48321229*n + 17452260)*a(n) = 72*(n-1)*n*(3*n - 5)*(3*n - 4)*(127401984*n^9 - 2045067264*n^8 + 14240360448*n^7 - 56278911936*n^6 + 138595592064*n^5 - 219567715966*n^4 + 222542820712*n^3 - 138190518059*n^2 + 47259501167*n - 6683489400)*a(n-1) - 48*(n-1)*(16307453952*n^12 - 351459606528*n^11 + 3428587929600*n^10 - 20001961205760*n^9 + 77643945578496*n^8 - 211031837008384*n^7 + 411217026027200*n^6 - 577827896836090*n^5 + 579810023200127*n^4 - 403994885007838*n^3 + 184802213339825*n^2 - 49548085570200*n + 5838168798000)*a(n-2) + 128*(2*n - 5)*(4*n - 11)*(4*n - 9)*(8*n - 23)*(8*n - 21)*(8*n - 19)*(8*n - 17)*(147456*n^6 - 1113600*n^5 + 3430080*n^4 - 5488810*n^3 + 4779029*n^2 - 2123685*n + 369600)*a(n-3). - Vaclav Kotesovec, Feb 22 2015
a(n) ~ (256/27)^n / n^(5/4) * (3^(1/4)*sqrt(EllipticK(1/sqrt(2)))/(2*Pi)^(3/4) - sqrt(3/(2*Pi))/n^(1/4) + (2/(3*Pi))^(1/4) / sqrt(EllipticK(1/sqrt(2)))/n^(1/2)), where EllipticK(1/sqrt(2)) = A093341 = GAMMA(1/4)^2/(4*(Pi)^(1/2)) = 1.85407467730137191843385... (= EllipticK[1/2] in Mathematica). - Vaclav Kotesovec, Feb 22 2015

A105372 Decimal expansion of Hypergeometric2F1[ -(1/4),3/4,1,1] = sqrt(Pi)/(Gamma[1/4]*Gamma[5/4]).

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Apr 02 2005

This constant appears in solution to an ODE considered in A104996, A104997.

			0.53935260118837935666793572235555273276586896544304013033994...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Andrei Gruzinov, Power of an axisymmetric pulsar, Physical Review Letters, Vol. 94, No. 2 (2005), 021101, preprint, arXiv:astro-ph/0407279, 2004.

Cf. A093341, A104996, A104997.

evalf(1/EllipticK(1/sqrt(2)),120); # Vaclav Kotesovec, Jun 15 2015

RealDigits[1/EllipticK[1/2],10,120][[1]] (* Vaclav Kotesovec, Jun 15 2015 *)

sqrt(Pi)/(gamma(1/4)*gamma(5/4)) \\ G. C. Greubel, Jan 09 2017

Hypergeometric2F1[ -(1/4), 3/4, 1, 1] = Sqrt[Pi]/(Gamma[1/4]*Gamma[5/4]).
From Vaclav Kotesovec, Jun 15 2015: (Start)
4*sqrt(Pi)/Gamma(1/4)^2.
1 / EllipticK(1/sqrt(2)) (Maple notation).
1 / EllipticK[1/2] (Mathematica notation).
(End)
Equals Product_{k>=1} (1 + (-1)^k/(2*k)). - Amiram Eldar, Aug 26 2020

Last digit corrected by Vaclav Kotesovec, Jun 15 2015

cp's OEIS Frontend

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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A175576 Decimal expansion Pi^(3/2)/Gamma(3/4)^2.

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A224268 Decimal expansion of Product_{n>=1} (1 - 1/(4n+1)^2).

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A064853 Decimal expansion of the Lemniscate constant.

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A163973 Decimal expansion of Van der Pauw's constant = Pi/log(2).

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A153396 G.f.: A(x) = F(xG(x)^3) where F(x) = G(x/F(x)^2) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)^2) = 1 + xG(x)^4 is the g.f. of A002293.