A249205 -id:A249205 - cp's OEIS frontend

A335415 Decimal expansion of Sum_{k>=0} 1/cosh(Pi*k).

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

Offset: 1

Vaclav Kotesovec, Jun 08 2020

			1.090170299508048113022668970279244293616858317440723649757932199702152...

Cf. A240964, A254445, A335414. Essentially the same as A249205.

evalf(Sum(1/cosh(Pi*k), k=0..infinity), 120);
evalf(1/2 + sqrt(Pi) / (2*GAMMA(3/4)^2), 120);

RealDigits[1/2 + Gamma[1/4]^2/(4*Pi^(3/2)), 10, 120][[1]]

PARI
```
suminf(k=0, 1/cosh(Pi*k))
```

Equals 1/2 + Gamma(1/4)^2 / (4*Pi^(3/2)).

Equals 1/2 + sqrt(Pi) / (2*Gamma(3/4)^2).

A249206 Decimal expansion of the logarithmic capacity of the unit equilateral triangle.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 23 2014

			0.421753934648426824238122958592773059107710633283...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.9 Integer Chebyshev constants, p. 268.

Steven Finch, Electrical capacitance [Cached copy, with permission of the author]

Cf. A073005, A249205, A249220.

k = (Sqrt[3]/(8*Pi^2))*Gamma[1/3]^3; RealDigits[k, 10, 102] // First

k = (sqrt(3)/(8*Pi^2))*Gamma(1/3)^3.

A249220 Decimal expansion of a constant related to the [dimensionless] electrical capacitance of the ring torus without hole (with unit circle radius).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 23 2014

			0.435345066268971927532148125596320824348...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.9 Integer Chebyshev constants, p. 268.

Steven Finch, Electrical capacitance [Cached copy, with permission of the author]

Cf. A249205, A249206.

evalf(int(1/BesselI(0,x)^2, x=0..infinity)/Pi, 50); # Vaclav Kotesovec, Oct 23 2014

k = (1/Pi)*NIntegrate[1/BesselI[0, t]^2, {t, 0, Infinity}, WorkingPrecision -> 100]; RealDigits[k] // First

k = (1/Pi)*integral_{0..infinity} 1/I_0(t)^2 dt, where I_0 is the zeroth modified Bessel function of the first kind.

The electrical capacitance is 4*k = 1.74138...

cp's OEIS Frontend

A335415 Decimal expansion of Sum_{k>=0} 1/cosh(Pi*k).

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A249206 Decimal expansion of the logarithmic capacity of the unit equilateral triangle.

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A249220 Decimal expansion of a constant related to the [dimensionless] electrical capacitance of the ring torus without hole (with unit circle radius).

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Maple

Mathematica

Formula