A068466 -id:A068466 - cp's OEIS frontend

A256930 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)(3/4)^(2*k).

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

Offset: 1

Jean-François Alcover, Apr 13 2015

			1.2036214036775919014128244060883195641815351691976781420672...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, p. 272, eq. (31).

Eric Weisstein's MathWorld, Riemann Zeta Function.
Wikipedia, Riemann zeta function.

Cf. A068466, A203130, A256929.

RealDigits[Log[3*Pi/(2*Sqrt[2])], 10, 107] // First

log(3*Pi/(2*sqrt(2))) \\ Michel Marcus, Apr 13 2015

Equals log(Gamma(1/4)*Gamma(7/4)).

Equals log(3*Pi/(2*sqrt(2))).

Name corrected by Amiram Eldar, Oct 12 2024

A257094 Decimal expansion of Gamma(9/4).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 16 2015

			1.13300309631934634747833911120864750093598990090002...

Eric Weisstein's MathWorld, Gamma Function.
Wikipedia, Particular values of the Gamma function.

Cf. A068466 (Gamma(1/4)), A068465 (3/4), A068467 (5/4), A203130 (7/4), A257095 (11/4).

RealDigits[Gamma[9/4], 10, 102] // First

gamma(9/4) \\ Michel Marcus, Apr 16 2015

Equals (5/16)*Gamma(1/4).

A257095 Decimal expansion of Gamma(11/4).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 16 2015

			1.6083594219855456592319415231637938164922515131418426772395311...

David H. Bailey and Simon Plouffe, Recognizing Numerical Constants, (1995).
Eric Weisstein's MathWorld, Gamma Function
Wikipedia, Particular values of the Gamma function

Cf. A068466 (Gamma(1/4)), A068465 (3/4), A068467 (5/4), A203130 (7/4), A257094 (9/4).

RealDigits[Gamma[11/4], 10, 104] // First

gamma(11/4) \\ Michel Marcus, Apr 16 2015

(21/16)*Pi*sqrt(2)/Gamma(1/4).

Also equals Integral_{0..infinity} t^(7/4)*exp(-t) dt.

A257406 Decimal expansion of Integral_{0..infinity} log(x)/cosh(x) dx (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 22 2015

			-0.5208856126019768910801877375794544390636383554462853499753755842...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Victor Adamchik, A Class of Logarithmic Integrals (1997) Proc. of ISSAC'97

Cf. A068466, A115252.

RealDigits[(Pi/2)*Log[4*Pi^3/Gamma[1/4]^4], 10, 103] // First
RealDigits[Integrate[-Log[x]/Cosh[x],{x,0,\[Infinity]}],10,120][[1]] (* Harvey P. Dale, Feb 05 2025 *)

(Pi/2)*log(4*Pi^3/gamma(1/4)^4) \\ Michel Marcus, Apr 22 2015

(Pi/2)*log(4*Pi^3/Gamma(1/4)^4).
Also equals 2*Integral_{0..1} (1/(x^2+1))*log(log(1/x)) dx.
Also equals 2*Integral_{Pi/4..Pi/2} log(log(tan(x))) dx.

A263809 Decimal expansion of C_{1/2}, a constant related to Kolmogorov's inequalities.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 27 2015

			2.78640785937135371836849252065073648531496243503123575794856326376...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.7 Riesz-Kolmogorov Constants, p. 474.

Burgess Davis, On Kolmogorov's Inequalities, Transactions of the American Mathematical Society Vol. 222 (Sep., 1976), pp. 179-192.

Cf. A068465, A068466, A093341, A242822.

RealDigits[Gamma[1/4]^2/(Pi*Gamma[3/4]^2), 10, 105] // First

gamma(1/4)^2/(Pi*gamma(3/4)^2) \\ Michel Marcus, Oct 27 2015

C_{1/2} = gamma(1/4)^2/(Pi*gamma(3/4)^2).
Equals (1/Pi^2)*(integral_{0..Pi} sqrt(csc(t)) dt)^2.
Also equals (8/Pi^2)*A093341^2.

A275322 Decimal expansion of AGM(1, sqrt(2))^2/Pi.

Original entry on oeis.org

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

Offset: 0

Dimitris Valianatos, Jul 23 2016

Conjecture: Equals Product_{n odd} (n/(n+2) if n == 1 (mod 4), (n+2)/n otherwise) = (1/3) * (5/3) * (5/7) * (9/7) * (9/11) * (13/11) * (13/15) * (17/15) * (17/19) * (21/19) * (21/23) * (25/23) * (25/27) * ...

			0.45694658104446362537496662254768...

G. C. Greubel, Table of n, a(n) for n = 0..10000
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. 21.

Cf. A053004 (AGM(1, sqrt(2))).

Cf. A000796, A002161, A068466, A212002.

SetDefaultRealField(RealField(100)); R:= RealField(); 8*Pi(R)^2/Gamma(1/4)^4; // G. C. Greubel, Oct 07 2018

evalf(GaussAGM(1,sqrt(2))^2/Pi,100); # Muniru A Asiru, Oct 08 2018

First@ RealDigits@ N[ArithmeticGeometricMean[1, Sqrt[2]]^2/Pi, 120] (* Michael De Vlieger, Jul 26 2016 *)

PARI
```
agm(1, sqrt(2)) ^ 2 / Pi
```

8*Pi^2/gamma(1/4)^4 \\ Altug Alkan, Oct 08 2018

Equals 8*Pi^2/Gamma(1/4)^4 = 4*Gamma(3/4)^2/Gamma(1/4)^2. - Vaclav Kotesovec, Sep 22 2016

A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jan 04 2023

			0.276427204245986573092639825616889946783740795...

John M. Campbell, Applications of a class of transformations of complex sequences, arXiv:2212.13305 [math.NT], 2022. See p. 6.

Cf. A000796, A001008, A002805, A002897, A016639, A016813, A063448, A068465, A068466, A091925, A203142, A203143.

Cf. A359532.

First[RealDigits[N[(Gamma[1/8]Gamma[3/8]/(Gamma[1/4]Gamma[3/4]))^2/(6Sqrt[2]Pi)-4Log[2]/Pi,100]]]

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8)/(Gamma(1/4)*Gamma(3/4)))^2/(6*sqrt(2)*Pi).

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8))^2/(12*sqrt(2)*Pi^3).

A371855 Decimal expansion of Integral_{x=-oo..oo} exp(-x^4) dx.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 09 2024

			1.8128049541109541559653425779338360014975838...

Cf. A002161, A068466, A068467.

Mathematica
```
RealDigits[2 Gamma[5/4], 10, 104][[1]]
```

Equals 2 * Gamma(5/4).
Equals 2 * A068467.
Equals A068466 / 2.

A240965 Decimal expansion of integral_(0..1) K(1-x^2)^3 dx, where K is the complete elliptic integral of the first kind.

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Aug 05 2014

			23.634090016154231536632674566865164174841395158861393288529...

M. Rogers, J. G. Wan, and I. J. Zucker, Moments of elliptic entegrals and critical L-values.
Eric Weisstein's MathWorld, Complete Elliptic Integral of the First Kind.

Cf. A068466.

(* NIntegrate[EllipticK[1 - x^2]^3, {x, 0, 1}] *)
RealDigits[Gamma[1/4]^8/(128*Pi^2), 10, 105] // First

intnum(x=0,1,ellK(sqrt(1-x^2))^3) \\ Charles R Greathouse IV, Feb 05 2025

gamma(1/4)^8/128/Pi^2 \\ Charles R Greathouse IV, Feb 05 2025

Gamma(1/4)^8/(128*Pi^2).

A246856 Decimal expansion of 'alpha', the threshold angle associated with the best constant in [a variation of] Hardy's inequality for a domain defined as a non-convex plane sector of angle alpha.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 05 2014

			4.85605532093166203862643534890529398551953783102088...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 30.

Cf. A068465, A068466.

alpha = Pi + 4*ArcTan[4*Gamma[3/4]^2/Gamma[1/4]^2]; RealDigits[alpha, 10, 102] // First

Pi + 4*atan(4*gamma(3/4)^2/gamma(1/4)^2) \\ Michel Marcus, Sep 05 2014

alpha = Pi + 4*arctan(4*Gamma(3/4)^2/Gamma(1/4)^2).

cp's OEIS Frontend

A256930 Decimal expansion of Sum_{k>=1} (zeta(2*k)/k)*(3/4)^(2*k).

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A257094 Decimal expansion of Gamma(9/4).

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A257095 Decimal expansion of Gamma(11/4).

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A257406 Decimal expansion of Integral_{0..infinity} log(x)/cosh(x) dx (negated).

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A263809 Decimal expansion of C_{1/2}, a constant related to Kolmogorov's inequalities.

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A275322 Decimal expansion of AGM(1, sqrt(2))^2/Pi.

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A359533 Decimal expansion of Sum_{k>=0} (-1/64)^k*binomial(2*k, k)^3*(4*k + 1)*H_k, where H_k is the k-th harmonic number (negated).

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A256930 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)(3/4)^(2*k).

A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).