A091670 -id:A091670 - 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

A277232 Numerators of the partial sums of the cubes of the expansion coefficients of 1/sqrt(1-x).

Original entry on oeis.org

1, 9, 603, 4949, 2576763, 20864151, 1347632055, 10860010029, 44749069441659, 359788384157147, 23124997294306677, 185685617347012755, 95380005326947177879, 765237422887515344907, 49101291379356533433423, 393721549706169405868509, 12928613856208967961607217787

Offset: 0

Wolfdieter Lang, Nov 11 2016

The denominators seem to coincide with A241756.
These are the partial sums of F. Morley's series Sum_{k>=0} (risefac(m,k)/k!)^3 for m=1/2, where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, pp. 104, 111.
The Morley formula gives the value of this series for |m| < 2/3 as Gamma(1-3*m/2)/(Gamma(1-m/2)^3)*cos(Pi*m/2). For the present case m=1/2 this value is hypergeometric([1/2,1/2,1/2],[1,1],1) = Pi/Gamma(3/4)^4 given in A091670.

			The rationals r(n) begin: 1, 9/8, 603/512, 4949/4096, 2576763/2097152, 20864151/16777216, 1347632055/1073741824, ...
The limit is given in A091670, approximately 1.3932039296856768591...

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 104.

Seiichi Manyama, Table of n, a(n) for n = 0..555
F. Morley, On the Series 1 + (p/1)^3 + {p*(p+1)/1.2}^3 + ... , Proc. London Math. Soc. 34 (1902) 397-402, eq. (5), p. 401.
Eric Weisstein's World of Mathematics, Morley's Formula.

Cf. A001147, A000165, A091670, A241756.

a(n) = numerator(r(n)) with the rational r(n) = Sum_{k=0..n} (risefac(1/2,k)/k!)^3 = Sum_{k=0..n} (-1)^k*(binomial(-1/2,k))^3 = Sum_{k=0..n} ((2*k-1)!!/(2*k)!!)^3. The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.

A293238 Decimal expansion of the escape probability for a random walk on the 3D bcc lattice.

Original entry on oeis.org

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

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, A091670.

			0.7177700110461299978211932236657794...

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 b.c.c. lattice.
Index entries for sequences related to walks.

Cf. A091670, A242761, A293237.

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

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

default(realprecision, 100); (4*Pi^3)/gamma(1/4)^4 \\ G. C. Greubel, Oct 26 2018

Equals Pi^2/(4*K(1/sqrt(2))^2), where K is the complete elliptic integral of the first kind.
Equals (4*Pi^3)/Gamma(1/4)^4. - G. C. Greubel, Oct 26 2018
Equals Product_{n>=1} exp(beta(2n)/n), where beta(n) is the Dirichlet beta function. - Antonio Graciá Llorente, Apr 03 2025
Equals Gamma(3/4)^4/Pi. - Stefano Spezia, Apr 05 2025

A378130 Decimal expansion of 24L^2/(5^(7/4)Pi^2), where L is the lemniscate constant (A062539).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 18 2024

			0.999996383159084127772763499184706112808943488770...

Wikipedia, Lemniscate constant (includes this constant).
Index to sequences related to the Gamma function.

Cf. A008977, A062539, A091670.

Cf. A377999, A378128, A378129, A378129, A378131, A378132.

First[RealDigits[12*Pi/(5^(7/4)*Gamma[3/4]^4), 10, 100]] (* or *)
First[RealDigits[Sum[((-1)^k/6635520^k)*(4*k)!/k!^4, {k, 0, Infinity}], 10, 100]] (* or *)
First[RealDigits[HypergeometricPFQ[{1/4, 1/2, 3/4}, {1, 1}, -1/25920], 10, 100]]

Equals 12*Pi/(5^(7/4)*Gamma(3/4)^4) = 12*A091670/5^(7/4).
Equals Sum_{k >= 0} ((-1)^k/6635520^k)*(4*k)!/(k!)^4 = Sum_{k >= 0} ((-1)^k/6635520^k)*A008977(k).
Equals pFq(1/4, 1/2, 3/4; 1, 1; -1/25920), where pFq is the generalized hypergeometric function.

A091671 Decimal expansion of (3Gamma(1/3)^6)/(162^(2/3)*Pi^4).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 27 2004

Watson's second triple integral.

			0.448220394388381432116385450017485249569392201708120730....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Watson's Triple Integrals

Cf. A091670, A091672.

SetDefaultRealField(RealField(100)); R:= RealField(); (3*Gamma(1/3)^6)/(16*2^(2/3)*Pi(R)^4); // G. C. Greubel, Oct 26 2018

RealDigits[(3*Gamma[1/3]^6)/(16*2^(2/3)*Pi^4), 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)

default(realprecision, 100); (3*gamma(1/3)^6)/(16*2^(2/3)*Pi^4) \\ G. C. Greubel, Oct 26 2018

A091672 Decimal expansion of (4(18+12sqrt(2)-10sqrt(3)-7sqrt(6)))EllipticK((2-sqrt(3))(-sqrt(2)+sqrt(3)))^2/Pi^2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 27 2004

Watson's third triple integral.

			0.505462019717326006052004053227140259985129014817420892188993487886...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, Ten Problems in Experimental Mathematics
Eric Weisstein's World of Mathematics, Watson's Triple Integrals

Cf. A091670, A091671.

evalf((4*(18+12*sqrt(2)-10*sqrt(3)-7*sqrt(6)))*EllipticK((2-sqrt(3))*(-sqrt(2)+sqrt(3)))^2/Pi^2, 120); # Vaclav Kotesovec, Apr 22 2015

RealDigits[ N[ (4*(18 + 12*Sqrt[2] - 10*Sqrt[3] - 7*Sqrt[6])*EllipticK[(2 - Sqrt[3])^2*(-Sqrt[2] + Sqrt[3])^2]^2)/Pi^2, 102]][[1]] (* Jean-François Alcover, Nov 12 2012, after Eric W. Weisstein *)

4*(18+12*sqrt(2)-10*sqrt(3)-7*sqrt(6))*ellK((2-sqrt(3))*(sqrt(3)-sqrt(2)))^2/Pi^2 \\ Charles R Greathouse IV, Feb 04 2025

Name corrected by Charles R Greathouse IV, Feb 04 2025

A277235 Decimal expansion of 2/(Gamma(3/4))^4.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 13 2016

This is the value of one of Ramanujan's series: 1 - 5*(1/2)^5 + 9*(1*3/(2*4))^5 -13*(1*3*5/(2*4*6))^5 + - ... . See the Hardy reference p.7. eq. (1.4) and pp. 105-106. For the partial sums see A278140.

The proof of Hardy and Whipple mentioned in the Hardy reference reduces this series to (2/Pi)*Morley's series (for m=1/2). For this series see A277232 and A091670.

			2/Gamma(3/4)^4 = 0.88694116857811540541152...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A014549, A060294, A074799, A074800, A088538, A091670, A263809, A277232, A278140.

SetDefaultRealField(RealField(100)); 2/(Gamma(3/4))^4; // G. C. Greubel, Oct 26 2018

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

2/gamma(3/4)^4 \\ Michel Marcus, Nov 13 2016

Equals Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^5 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^5. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
Equals A060294 * A091670.
For (1+4*k)*((2*k-1)!!/(2*k)!!)^5 see A074799(k) / A074800(k).
From Amiram Eldar, Jul 13 2023: (Start)
Equals (Gamma(1/4)/Pi)^4/2.
Equals A088538 * A014549^2.
Equals A263809/Pi. (End)

A352615 Decimal expansion of Integral_{0<=x,y<=Pi/2} sqrt(1-cos^2(x)*cos^2(y)) dx dy.

Original entry on oeis.org

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

Offset: 1

Robert FERREOL, Mar 23 2022

			2.0776814600281582057831205567855290128037790576247...

Robert Ferréol, Bohemian dome, Mathcurve.

Cf. A091670 ((1/Pi^2)*Integral_{0<=x,y<=Pi} 1/sqrt(1-cos^2(x)*cos^2(y)) dx dy).

a:=1/sqrt(2):evalf((EllipticE(a)-EllipticK(a))^2+EllipticE(a)^2,50);

RealDigits[(EllipticE[1/2] - EllipticK[1/2])^2 + EllipticE[1/2]^2, 10, 105][[1]] (* Amiram Eldar, Mar 24 2022 *)

Equals Sum _{n>=0} (Pi^2/(4*(2*n-1))*(binomial(2*n,n)/4^n)^3).

Equals (E(a) - K(a))^2 + E(a)^2 where a = 1/sqrt(2) and E (resp. K) is the complete elliptic integral of the second (resp. first) kind.

cp's OEIS Frontend

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

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A277232 Numerators of the partial sums of the cubes of the expansion coefficients of 1/sqrt(1-x).

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A293238 Decimal expansion of the escape probability for a random walk on the 3D bcc lattice.

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A378130 Decimal expansion of 24*L^2/(5^(7/4)*Pi^2), where L is the lemniscate constant (A062539).

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A091671 Decimal expansion of (3*Gamma(1/3)^6)/(16*2^(2/3)*Pi^4).

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A091672 Decimal expansion of (4*(18+12*sqrt(2)-10*sqrt(3)-7*sqrt(6)))*EllipticK((2-sqrt(3))*(-sqrt(2)+sqrt(3)))^2/Pi^2.

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A277235 Decimal expansion of 2/(Gamma(3/4))^4.

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A378130 Decimal expansion of 24L^2/(5^(7/4)Pi^2), where L is the lemniscate constant (A062539).

A091671 Decimal expansion of (3Gamma(1/3)^6)/(162^(2/3)*Pi^4).

A091672 Decimal expansion of (4(18+12sqrt(2)-10sqrt(3)-7sqrt(6)))EllipticK((2-sqrt(3))(-sqrt(2)+sqrt(3)))^2/Pi^2.