A085565 -id:A085565 - cp's OEIS frontend

A175574 Decimal expansion of sqrt(Pi) / (Gamma(3/4))^2.

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

Offset: 1

R. J. Mathar, Jul 15 2010

Entry 34 c of chapter 11 of Ramanujan's second notebook.

This constant is also the ratio T(Pi/2)/T(0), where T(Pi/2) is the exact pendulum period for an amplitude of Pi/2 and T(0) the approximate period 2*Pi*sqrt(L/g) for small angles. - Jean-François Alcover, Aug 05 2014

			1.18034059901609622604533794..

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.
Claudio Carvalhaes and Patrick Suppes, Approximations for the period of the simple pendulum based on the arithmetic-geometric mean, American Journal of Physics 76 (2008), 1150-1154.

Cf. A002161, A068465, A085565, A096427, A175573.

sqrt(pi)/gamma(3/4)^2 % Altug Alkan, Dec 05 2015

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

First@ RealDigits[N[Sqrt@ Pi/Gamma[3/4]^2, 120]] (* Michael De Vlieger, Dec 06 2015 *)

sqrt(Pi)/gamma(3/4)^2 \\ Altug Alkan, Dec 05 2015

Equals A002161 /A068465^2.
Equals 2F1([1/2,1/2],[1],1/2) = 1/agm(1, sqrt(1/2)) = gamma(1/4)^2/(2*Pi^(3/2)).
Equals 2*sqrt(2)*K(-1)/Pi, where K is the complete elliptic integral of the first kind, K(-1) being A085565. - Jean-François Alcover, Jun 03 2014
Equals Product_{k>=1} (1-(-1)^k/(2*k)) =  3/2 * 3/4 * 7/6 * 7/8 * 11/10 * 11/12 * ... .  - Richard R. Forberg, Dec 05 2015
Reciprocal of A096427. Equals ( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2, a rapidly converging series. For example, summing from n = -5 to n = 5 gives the constant correct to 49 decimal places. - Peter Bala, Mar 06 2019
Equals Sum_{k>=0} binomial(2*k,k)^2/2^(5*k). - Amiram Eldar, Aug 26 2020
Equals (3/2)*hypergeom([-1/4, 3/4], [3/2], 1). - Peter Bala, Mar 04 2022
Equals A175573^2. - Amiram Eldar, Jul 04 2023

A-number typo for sqrt(Pi) corrected by R. J. Mathar, Aug 01 2010

A113477 Decimal expansion of Gamma(1/3)^3/(2^(4/3)*Pi).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jan 08 2006

This number is transcendental from a result of Schneider on elliptic integrals.

			2.428650647887581611819....

G. C. Greubel, Table of n, a(n) for n = 0..5000
Th. Schneider, Transzendenzuntersuchungen periodischer Funktionen, Journal für die reine und angewandte Mathematik (1935) Volume: 172, page 65-74.
Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale, Mathematische Annalen (1937) Volume: 113, page I-XIII.
Index entries for transcendental numbers

Cf. A085565.

Beta(1/6, 1/2)/3: evalf(%, 106); # Peter Luschny, Apr 15 2024

RealDigits[Gamma[1/3]^3/(Pi*2^(4/3)), 10, 5001][[1]] (* G. C. Greubel, Mar 12 2017 *)

PARI
```
gamma(1/3)^3/2^(4/3)/Pi
```

Equals Integral_{x>=1} dx/sqrt(4*x^3-4).
Equals 2*Integral_{x=0..1} dx/sqrt(1-x^6). - Takayuki Tatekawa, Apr 15 2024
Equals Beta(1/6, 1/2) / 3. - Peter Luschny, Apr 15 2024

A154743 Decimal expansion of 2^(1/4) - 2^(-1/4), the ordinate of the point of bisection of the arc of the unit lemniscate (x^2 + y^2)^2 = x^2 - y^2 in the first quadrant.

Original entry on oeis.org

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

Offset: 0

Stuart Clary, Jan 14 2009

A quartic integer with denominator 2: the positive root of 2x^4 + 8x^2 - 1 = 0.

			0.348310699749006523686374494327...

C. L. Siegel, Topics in Complex Function Theory, Volume I: Elliptic Functions and Uniformization Theory, Wiley-Interscience, 1969, page 5

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for algebraic numbers, degree 4

Cf. A154739 for the abscissa and A154747 for the radius vector.
Cf. A154744, A154745 and A154746 for the continued fraction and the numerators and denominators of the convergents.
Cf. A085565 for 1.311028777..., the first-quadrant arc length of the unit lemniscate.

[2^(1/4) - 2^(-1/4)]; // G. C. Greubel, Nov 05 2017

nmax = 1000; First[ RealDigits[ 2^(1/4) - 2^(-1/4), 10, nmax] ]

sqrtn(2, 4) - sqrtn(2, -4) \\ Michel Marcus, Dec 10 2016

polrootsreal(2*x^4+8*x^2-1)[2] \\ Charles R Greathouse IV, Nov 07 2017

Offset corrected by R. J. Mathar, Feb 05 2009

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

A371824 Decimal expansion of Pi^(1/2)Gamma(1/10)/(5Gamma(3/5)).

Original entry on oeis.org

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

Offset: 1

Takayuki Tatekawa, Apr 07 2024

Constants from generalized Pi integrals: the case of n=10.

			2.264617395043150744291188990313...

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427.

RealDigits[2*Sqrt[Pi]/10*Gamma[1/10]/Gamma[3/5], 10, 5001][[1]]
RealDigits[GoldenRatio * Gamma[1/5] * Gamma[2/5]^2 / (2^(6/5) * Sqrt[5] * Pi), 10, 120][[1]] (* Vaclav Kotesovec, Apr 07 2024 *)

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^10).

Equals phi * Gamma(1/5) * Gamma(2/5)^2 / (2^(6/5) * sqrt(5) * Pi), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Apr 07 2024

A378128 Decimal expansion of 2/L, where L is the lemniscate constant (A062539).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 17 2024

			0.76275976350181318806232598096361579329262923734807...

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

Cf. A010466, A062539, A085565, A175575.

Cf. A377999, A378129, A378130, A378131, A378132.

First[RealDigits[Sqrt[8]*Gamma[3/4]^2/Pi^(3/2), 10, 100]]

Equals 1/A085565.
Equals 2*sqrt(2)*Gamma(3/4)^2/Pi^(3/2) = A010466*A175575.
Equals Product_{k >= 1} b(k), where b(1) = sqrt(1/2) and, for k >= 2, b(k) = sqrt(1/2 + (1/2)/b(k-1)).

A243340 Decimal expansion of 4L/(3Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 03 2014

			1.11283578889876424837523964373206241199199...

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 102.

Eric Weisstein's World of Mathematics, Lemniscate Constant.

Cf. A062539 (L), A076390, A085565, A225119 (L/3).

L = Pi^(3/2)/(Sqrt[2]*Gamma[3/4]^2); RealDigits[4*L/(3*Pi), 10, 103] // First

2*sqrt(2*Pi)/(3*gamma(3/4)^2) \\ Stefano Spezia, Nov 27 2024

Equals 2*sqrt(2*Pi)/(3*Gamma(3/4)^2).
From  Peter Bala, Mar 24 2024: (Start)
An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 3 for k >= 0.
For example, taking k = 0 and k = 1 yields
4*L/(3*Pi) = 1 + 1/(6 + (5*7)/(6 + (9*11)/(6 + (13*15)/(6 + ... + (4*n + 1)*(4*n + 3)/(6 + ... ))))) and
4*L/(3*Pi) = 8/(7 + (1*3)/(14 + (5*7)/(14 + (9*11)/(14 + (13*15)/(14 + ... + (4*n + 1)*(4*n + 3)/(14 + ... )))))).
Equals (2/3) * 1/A076390. (End)

A371930 Decimal expansion of Pi^(1/2)Gamma(1/14)/(7Gamma(4/7)).

Original entry on oeis.org

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

Offset: 1

Takayuki Tatekawa, Apr 12 2024

Constants from generalized Pi integrals: the case of n=14.

In general, for k > 0, Integral_{x=0..1} 1/sqrt(1 - x^k) dx = 2^(2/k) * Gamma(1 + 1/k)^2 / Gamma(1 + 2/k) = 2^(2/k - 1) * Gamma(1/k)^2 / (k*Gamma(2/k)). - Vaclav Kotesovec, Apr 15 2024

			2.191450245201078533946264870311...

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427, A371824.

Beta(1/14, 1/2) / 7: evalf(%, 90); # Peter Luschny, Apr 14 2024

RealDigits[Sqrt[Pi]/7*Gamma[1/14]/Gamma[4/7], 10, 5001][[1]]

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^14).
Equals Beta(1/14, 1/2) / 7. - Peter Luschny, Apr 14 2024
Equals Gamma(1/14)^2 / (7 * 2^(6/7) * Gamma(1/7)). - Vaclav Kotesovec, Apr 15 2024

A371929 Decimal expansion of Pi^(1/2)Gamma(1/12)/(6Gamma(7/12)).

Original entry on oeis.org

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

Offset: 1

Takayuki Tatekawa, Apr 12 2024

Constants from generalized Pi integrals: the case of n=12.

			2.2221586039664144669155853439....

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427, A371824.

Beta(1/12, 1/2) / 6: evalf(%, 89); # Peter Luschny, Apr 14 2024

RealDigits[Sqrt[Pi]/6*Gamma[1/12]/Gamma[7/12], 10, 5001][[1]]
RealDigits[(1 + Sqrt[3]) * Gamma[1/4]^2 / (4 * 3^(3/4) * Sqrt[Pi]), 10, 120][[1]] (* Vaclav Kotesovec, Apr 15 2024 *)

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^12).
Equals Beta(1/12, 1/2) / 6. - Peter Luschny, Apr 14 2024
Equals (1 + sqrt(3)) * Gamma(1/4)^2 / (4 * 3^(3/4) * sqrt(Pi)). - Vaclav Kotesovec, Apr 15 2024

A372327 Decimal expansion of Pi^(1/2)Gamma(1/18)/(9Gamma(5/9)).

Original entry on oeis.org

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

Offset: 1

Takayuki Tatekawa, Apr 28 2024

Constant from generalized Pi integrals: the case of n=18.

			2.14999545849204723391222945664...

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427, A371929, A371930.

RealDigits[Sqrt[Pi]/9*Gamma[1/18]/Gamma[5/9], 10, 5001][[1]]

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^18).

Equals Gamma(1/18)^2 / (9 * 2^(8/9) * Gamma(1/9)). - Vaclav Kotesovec, Apr 29 2024

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A175574 Decimal expansion of sqrt(Pi) / (Gamma(3/4))^2.

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A113477 Decimal expansion of Gamma(1/3)^3/(2^(4/3)*Pi).

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A154743 Decimal expansion of 2^(1/4) - 2^(-1/4), the ordinate of the point of bisection of the arc of the unit lemniscate (x^2 + y^2)^2 = x^2 - y^2 in the first quadrant.

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

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A371824 Decimal expansion of Pi^(1/2)*Gamma(1/10)/(5*Gamma(3/5)).

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A378128 Decimal expansion of 2/L, where L is the lemniscate constant (A062539).

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A243340 Decimal expansion of 4*L/(3*Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.

A371824 Decimal expansion of Pi^(1/2)Gamma(1/10)/(5Gamma(3/5)).

A243340 Decimal expansion of 4L/(3Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant.

A371930 Decimal expansion of Pi^(1/2)Gamma(1/14)/(7Gamma(4/7)).

A371929 Decimal expansion of Pi^(1/2)Gamma(1/12)/(6Gamma(7/12)).

A372327 Decimal expansion of Pi^(1/2)Gamma(1/18)/(9Gamma(5/9)).