A175574 -id:A175574 - cp's OEIS frontend

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

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

A091670 Decimal expansion of Gamma(1/4)^4/(4*Pi^3).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 27 2004

Watson's first triple integral.
This is also the value of F. Morley's series from 1902 Sum_{k=0..n} (risefac(k,1/2)/k!)^3 = hypergeometric([1/2,1/2,1/2],[1,1],1) with the rising factorial risefac(n,x). See A277232, also for the Hardy reference and a MathWorld link. - Wolfdieter Lang, Nov 11 2016
This constant is transcendental due to a result of Nesterenko, who proves that Gamma(1/4) is algebraically independent of Pi. - Charles R Greathouse IV, Aug 19 2025

			1.39320392968567685918424626032536824265748121751561787897...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 324.

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 256, 6.1.17 , p. 557, 15.1.26.
M. L. Glasser, I. J. Zucker, Extended Watson integrals for the cubic lattices, Proc. Nat. Acad. Sci., Vol. 74, No. 5 (1977), p. 1800-1801.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.5.1)
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics, Vol. 187, No. 9 (1996), pp. 1319-1348. (English translation)
Tito Piezas III, Watson's triple integrals.
Eric Weisstein's World of Mathematics, Watson's Triple Integrals.
I. J. Zucker, 70+years of the Watson integrals, J. Stat. Phys., Vol. 145, No. 3 (2011), pp. 591-612.
Index entries for transcendental numbers.

Cf. A091671, A091672, A277232, A293238 (inverse), A068466.

SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^4/(4*Pi(R)^3); // G. C. Greubel, Oct 26 2018

Pi/GAMMA(3/4)^4 ; evalf(%) ; # R. J. Mathar, Jun 17 2016

RealDigits[ N[ Gamma[1/4]^4/(4*Pi^3), 102]][[1]] (* Jean-François Alcover, Nov 12 2012, after Eric W. Weisstein *)

1/agm(sqrt(1/2),1)^2 \\ Charles R Greathouse IV, Mar 03 2016

From Joerg Arndt, Nov 27 2010: (Start)
Equals 1/agm(1,sqrt(1/2))^2.
Equals Gamma(1/4)^4 / (4*Pi^3) = Pi / (Gamma(3/4))^4 = hypergeom([1/2,1/2],[1],1/2)^2, see the two Abramowitz - Stegun references. (End)
Equals the square of A175574. Equals A000796/A068465^4. - R. J. Mathar, Jun 17 2016
Equals hypergeom([1/2,1/2,1/2],[1,1],1) - Wolfdieter Lang, Nov 12 2016
Equals Sum_{k>=0} binomial(2*k,k)^3/2^(6*k). - Amiram Eldar, Aug 26 2020

A243308 Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 03 2014

			1.0174087975959560086695387533500634259952569...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.4 Gamma function, p. 34.

Eric Weisstein's MathWorld, Gamma function

Cf. A073005, A175379, A175574.

Re(evalf(4*EllipticK(sqrt((4*sqrt(3)-7)))/(sqrt(2+sqrt(3))*Pi), 120)); # Vaclav Kotesovec, Apr 22 2015

RealDigits[4*EllipticK[4*Sqrt[3]-7]/(Sqrt[2+Sqrt[3]]*Pi), 10, 103] // First
RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2 + Sqrt[3]]/2], 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)
RealDigits[2 EllipticK[(2 - Sqrt[3])/4]/Pi, 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)

4*K(4*sqrt(3)-7)/(sqrt(2+sqrt(3))*Pi), where K is the complete elliptic integral of the first kind.
3^(1/4)*GAMMA(1/3)^3/(2*2^(1/3)*Pi^2), where GAMMA is the Euler Gamma function.
GAMMA(1/6)^(3/2)/(2^(5/6)*sqrt(3)*Pi^(5/4)).

A256514 Decimal expansion of the amplitude of a simple pendulum the period of which is twice the period in the small-amplitude approximation.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 01 2015

			2.7882311241072043014215218475308907276159087254649493...
= 159.75387571836004625994511811959034206912586138415864587... in degrees.

Claudio Carvalhaes and Patrick Suppes, Approximations for the period of the simple pendulum based on the arithmetic-geometric mean, American Journal of Physics 76, 1150-1154 (2008).
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean.
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.
Wikipedia, Pendulum.

Cf. A175574, A226204.

a2 = a /. FindRoot[ (2*EllipticK[ Sin[a/2]^2 ])/Pi == 2, {a, 3}, WorkingPrecision -> 102]; RealDigits[a2] // First

solve(x=2,3,1/agm(cos(x/2),1)-2) \\ Charles R Greathouse IV, Mar 03 2016

Solution to (2*K(sin(a/2)^2))/Pi = 2, where K is the complete elliptic integral of the first kind.

Also solution to 1/AGM(1, cos(a/2)) = 2, where AGM is the arithmetic-geometric mean.

A376643 Decimal expansion 4*EllipticK(4/5)/sqrt(5), where EllipticK is the complete elliptic integral of the first kind.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 01 2024

A point mass is attached to a frictionless pivot by a massless string of length L and revolves in a vertical circle about the pivot in a uniform gravitational field with an acceleration g. The slowest possible motion occurs when the tension in the string is momentarily zero at the top of the route, and the longest-possible period is then c * sqrt(L/g), where c is this constant.

			4.03781163995684643116802887999786493013608399340880...

Physics StackExchange, Maximum period of a vertically spinning ball, 2016.
Physics StackExchange, Maximal time for an object vertically swinging on a rope to complete a full circle, 2023.
Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.
Wikipedia, Elliptic integral: Complete elliptic integral of the first kind.

Constants related to similar physical problems: A019692, A038533, A038534, A175574, A256514, A309893, A310000.

RealDigits[4 * EllipticK[4/5] / Sqrt[5], 10, 120][[1]]

PARI
```
4*ellK(sqrt(4/5))/sqrt(5)
```

Equals 2 * Integral_{0..Pi} (1/sqrt(3 + 2*cos(x))) dx.

cp's OEIS Frontend

A175573 Decimal expansion of Pi^(1/4)/Gamma(3/4).

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

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A091670 Decimal expansion of Gamma(1/4)^4/(4*Pi^3).

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A243308 Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.

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A256514 Decimal expansion of the amplitude of a simple pendulum the period of which is twice the period in the small-amplitude approximation.

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A376643 Decimal expansion 4*EllipticK(4/5)/sqrt(5), where EllipticK is the complete elliptic integral of the first kind.

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