A293238 -id:A293238 - 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

A242761 Decimal expansion of the escape probability for a random walk on the 3-D cubic lattice (a Polya random walk constant).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 22 2014

			0.6594626704490008571737268155670971...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9, p. 322.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Polya's Random Walk Constants

Cf. A086230, A086231, A086232-A086236, A043546, A293237, A293238, A242812-A242816.

SetDefaultRealField(RealField(100)); R:= RealField(); (16*Sqrt(2/3)*Pi(R)^3)/(Gamma(1/24)*Gamma(5/24)*Gamma(7/24)*Gamma(11/24)); // G. C. Greubel, Oct 26 2018

p = (16*Sqrt[2/3]*Pi^3)/(Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]); RealDigits[p, 10, 100] // First

default(realprecision, 100); (16*sqrt(2/3)*Pi^3)/(gamma(1/24)* gamma(5/24)*gamma(7/24)*gamma(11/24)) \\ G. C. Greubel, Oct 26 2018

Equals (16*sqrt(2/3)*Pi^3)/(Gamma(1/24)*Gamma(5/24)*Gamma(7/24)*Gamma(11/24)), where Gamma is the Euler Gamma function.

A293237 Decimal expansion of the escape probability for a random walk on the 3D fcc lattice.

Original entry on oeis.org

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

Offset: 0

Andrey Zabolotskiy, Oct 03 2017

The return probability equals unity minus this constant. The expected number of visits to the origin is the inverse of this constant.

The escape probability for the hcp lattice also equals this constant. The escape probability for the diamond lattice is 3/4 times this constant.

			0.74368176349535122890496981936537648...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Shunya Ishioka and Masahiro Koiwa, Random walks on diamond and hexagonal close packed lattices, Phil. Mag. A, 37 (1978), 517-533.
G. L. Montet, Integral methods in the calculation of correlation factors in diffusion, Phys. Rev. B 7 (1973), 650-662.
Index entries for sequences related to f.c.c. lattice
Index entries for sequences related to walks

Cf. A242761, A293238.

SetDefaultRealField(RealField(100)); R:= RealField(); 2^(14/3)*Pi(R)^4/(9*Gamma(1/3)^6); // G. C. Greubel, Oct 26 2018

RealDigits[2^(14/3)*Pi^4/(9*Gamma[1/3]^6), 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)

2^(14/3)*Pi^4/(9*gamma(1/3)^6) \\ Altug Alkan, Apr 09 2018

Equals 2^(14/3)*Pi^4/(9*Gamma(1/3)^6).

cp's OEIS Frontend

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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A242761 Decimal expansion of the escape probability for a random walk on the 3-D cubic lattice (a Polya random walk constant).

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A293237 Decimal expansion of the escape probability for a random walk on the 3D fcc lattice.

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