A245058 Decimal expansion of the real part of Li_2(I), negated.

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

Offset

0,1

Comments

This is the decimal expansion of the real part of the dilogarithm of the square root of -1. The imaginary part is Catalan's number (A006752).

5*Pi^2/24 = 10 * (this constant) equals the asymptotic mean of the abundancy index of the even numbers. - Amiram Eldar, May 12 2023

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..104 from Robert G. Wilson v)

R. J. Mathar, Some definite integrals over a power multiplied by four modified Bessel functions vixra:1606.0141 (2016) eq. (39)

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.3.13)

Formula

Also equals -zeta(2)/8 = -Pi^2/48.

Also equals the Bessel moment Integral_{0..inf} x I_1(x) K_0(x)^2 K_1(x) dx. - Jean-François Alcover, Jun 05 2016

From Terry D. Grant, Sep 11 2016: (Start)

Equals Sum_{n>=0} (-1)^n/(2n+2)^2.

Equals (Sum_{n>=1} 1/(2n)^2)/2 = A222171/2. (End)

Equals Sum_{k>=1} A007949(k)/k^2. - Amiram Eldar, Jul 13 2020

Equals a tenth of integral_0^{pi/2} arccos[cos x/(1+2 cos x)]dx [Nahin]. - R. J. Mathar, May 22 2024

Example

0.2056167583560283045590518958307531486523687376508498047169447786712509338004...

Mathematica

RealDigits[ Re[ PolyLog[2, I]], 10, 111][[1]] (* or *) RealDigits[ Zeta[2]/8, 10, 111][[1]] (* or *) RealDigits[ Pi^2/48, 10, 111][[1]]

Prog

(PARI) zeta(2)/8 \\ Charles R Greathouse IV, Aug 27 2014
(Sage)
(pi**2/48).n(200) # F. Chapoton, Mar 16 2020
(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); Pi(R)^2/48; // G. C. Greubel, Aug 25 2018

Crossrefs

Cf. A006752, A007949, A072691, A222171, A242301.

Sequence in context: A155524 A051111 A068558 * A240657 A262420 A240662

Adjacent sequences: A245055 A245056 A245057 * A245059 A245060 A245061

Keyword

nonn,cons

Author

Robert G. Wilson v, Aug 21 2014

Status

approved