A257870 -id:A257870 - cp's OEIS frontend

A257872 Decimal expansion of the Madelung type constant C(4|1) (negated).

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

Offset: 1

Jean-François Alcover, May 11 2015

Without sign, this is the volume of the intersection of the three (solid) hyperboloids x^2 + y^2 - z^2 <= 1; y^2 + z^2 - x^2 <= 1; z^2 + x^2 - y^2 <= 1. See Villarino et al. - Michel Marcus, Aug 12 2021

In other words, decimal expansion of the volume of the unit trihyperboloid. - Eric W. Weisstein, Sep 18 2021

			-5.54517744447956247533785697166541254460400107488204203296544...

Hassan Chamati and Nicholay S. Tonchev, Exact results for some Madelung type constants in the finite-size scaling theory, arXiv:cond-mat/0003235 [cond-mat.stat-mech], 2000.
Mark B. Villarino and Joseph C. Várilly, Archimedes' Revenge, arXiv:2108.05195 [math.HO], 2021.
Eric Weisstein's World of Mathematics, Madelung Constants
Eric Weisstein's World of Mathematics, Trihyperboloid
Index entries for transcendental numbers

Cf. A000045, A257870, A257871.

Cf. A347903 (surface area of the unit trihyperboloid).

evalf(-8*log(2),120); # Vaclav Kotesovec, May 11 2015

Mathematica
```
RealDigits[-8*Log[2], 10, 103] // First
```

-8*log(2) \\ Charles R Greathouse IV, Sep 02 2021

-8*log(2).

4*log(2)/5 = 8*log(2)/10 = Sum_{k>=1} F(k)^2/(k * 3^k), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Aug 09 2020

A257871 Decimal expansion of the Madelung type constant C(2|1/2) (negated).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 11 2015

			-6.913039577009161107850187814269779123021008950691594327139798329827...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Hassan Chamati and Nicholay S. Tonchev, Exact results for some Madelung type constants in the finite-size scaling theory, arXiv:cond-mat/0003235 [cond-mat.stat-mech], 2000.
Eric Weisstein's World of Mathematics, Madelung Constants

Cf. A257870, A257872.

evalf(2*sqrt(Pi)*Zeta(1/2)*(Zeta(0, 1/2, 1/4)-Zeta(0, 1/2, 3/4)), 120); # Vaclav Kotesovec, May 11 2015

RealDigits[2*Sqrt[Pi]*Zeta[1/2]*(Zeta[1/2, 1/4] - Zeta[1/2, 3/4]), 10, 104] // First

2*sqrt(Pi)*zeta(1/2)*(zetahurwitz(1/2, 1/4) - zetahurwitz(1/2, 3/4)) \\ Charles R Greathouse IV, Jan 31 2018

Equals 2*sqrt(Pi)*zeta(1/2)*(zeta(1/2, 1/4) - zeta(1/2, 3/4)).

Equals 4*Pi^(1 - 2*nu)*gamma(nu)*zeta(nu)*DirichletBeta(nu) with nu = 1/2.

A365281 Decimal expansion of the least real solution x > 0 of Gamma(1/4 + x/2)/(Pi^x*Gamma(1/4 - x/2)) = 1.

Original entry on oeis.org

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

Offset: 2

Thomas Scheuerle, Aug 31 2023

			18.56775084706966207279145836562344730...

Cf. A059750, A114720, A257870.

FindRoot[-1 + Gamma[1/4 - x/2]/(Pi^(-x) Gamma[1/4 + x/2]) == 0, {x, 18.5569, 18.5739}, WorkingPrecision -> 100]

Let x be this constant:
Gamma(1/4 - x/2)/(Pi^x*Gamma(1/4 + x/2)) = 1.
zeta((1/2) + x) = zeta((1/2) - x), where zeta is the Riemann zeta function.
(2*Pi)^(-1/2 - x)*(2*x - 1)*cos(Pi/4 + (Pi*x)/2)*Gamma(x - 1/2) = 1.
2^(1/2 - x)*Pi^(-1/2 - x)*sin(Pi/2 - (Pi*x)/2)*Gamma(1/2 + x) = 1.
Z(i*x) = -zeta((1/2) + x) = -zeta((1/2) - x), where Z is the Riemann-Siegel Z function and i is the imaginary unit. From this follows that theta(i*-x) = theta(i*x) is an odd multiple of Pi where theta is the Riemann-Siegel theta function. This can also be seen if we consider Hardy's definition of the Z function: Z(s) = Pi^(-i*s/2)*zeta((1/2) + i*s)*Gamma((1/4)+(i*s/2))^(1/2)/Gamma((1/4) - (i*s/2))^(1/2).

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A257872 Decimal expansion of the Madelung type constant C(4|1) (negated).

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A257871 Decimal expansion of the Madelung type constant C(2|1/2) (negated).

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A365281 Decimal expansion of the least real solution x > 0 of Gamma(1/4 + x/2)/(Pi^x*Gamma(1/4 - x/2)) = 1.

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