This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A186706 #61 Jun 06 2025 00:28:09
%S A186706 3,6,2,7,5,9,8,7,2,8,4,6,8,4,3,5,7,0,1,1,8,8,1,5,6,5,1,5,2,8,4,3,1,1,
%T A186706 4,6,4,5,6,8,1,3,2,4,9,6,1,8,5,4,8,1,1,5,1,1,3,9,7,6,9,8,7,0,7,7,6,2,
%U A186706 4,6,3,6,2,2,5,2,7,0,7,7,6,7,3,6,8,2,4,9,9,7,6,4,2,4,1,2,0,3,3,7,7,1,2,4,4
%N A186706 Decimal expansion of the Integral of Dedekind Eta(x*I) from x = 0..oo.
%C A186706 Reduction of the integral by _Robert Israel_, Jul 25 2012: (Start)
%C A186706 Use the definition of DedekindEta as a sum:
%C A186706 Eta(i*x) = Sum_{n=-oo..oo} (-1)^n*exp(-Pi*x*(6n-1)^2/12).
%C A186706 Now Integral_{x=0..oo} exp(-Pi*x*(6n-1)^2/12) dx = 12/(Pi*(6n-1)^2).
%C A186706 According to Maple, Sum_{n=-oo..oo} (-1)^n*12/(Pi*(6n-1)^2) is
%C A186706 2*3^(1/2)*(dilog(1-(1/2)*i-(1/2)*3^(1/2)) - dilog(1-(1/2)*i+(1/2)*3^(1/2)) - dilog(1+(1/2)*i+(1/2)*3^(1/2)) + dilog(1+(1/2)*i-(1/2)*3^(1/2)))/Pi
%C A186706 (Jonquiere's inversion formula -- see http://en.wikipedia.org/wiki/Polylogarithm)
%C A186706 (but note that Maple's dilog(z) is L_2(1-z) in the notation there) gives
%C A186706 dilog(1-(1/2)*i-(1/2)*3^(1/2)) + dilog(1+(1/2)*i-(1/2)*3^(1/2)) = (13/72)*Pi^2
%C A186706 and
%C A186706 dilog(1-(1/2)*i+(1/2)*3^(1/2)) + dilog(1+(1/2)*i+(1/2)*3^(1/2)) = -11*Pi^2/72
%C A186706 which give the desired multiple of Pi. (End)
%C A186706 Ratio of surface area of a sphere to the regular octahedron whose edge equals the radius of the sphere. - _Omar E. Pol_, Dec 30 2023
%D A186706 Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.
%H A186706 D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting series involving the central binomial coefficient</a>, Am. Math. Monthly 92 (7) (1985) 449.
%H A186706 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See p. 511.
%H A186706 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>.
%F A186706 Equals 2*Pi/sqrt(3), 2 times A093602, and in consequence equal to Sum_{m>=1} 3^m/(m*binomial(2m,m)) according to Lehmer. - _R. J. Mathar_, Jul 24 2012
%F A186706 Also equals Gamma(1/3)*Gamma(2/3) = A073005 * A073006. - _Jean-François Alcover_, Nov 24 2014
%F A186706 From _Amiram Eldar_, Aug 06 2020: (Start)
%F A186706 Equals Integral_{x=0..oo} log(1 + 1/x^3) dx.
%F A186706 Equals Integral_{x=-oo..oo} exp(x/3)/(exp(x) + 1) dx. (End)
%F A186706 Equals Integral_{x=0..2*Pi} 1/(2 + sin(x)) dx; since for a>1: Integral_{x=0..2*Pi} 1/(a + sin(x)) dx = 2*Pi/sqrt(a^2-1). - _Bernard Schott_, Feb 18 2023
%F A186706 Equals 4*A093766. - _Omar E. Pol_, Dec 30 2023
%F A186706 From _Stefano Spezia_, Jun 05 2025: (Start)
%F A186706 Equals Beta(1/3,2/3).
%F A186706 Equals Integral_{x=-oo..oo} 1/(x^2 + x + 1) dx.
%F A186706 Equals 2*Integral_{x=0..oo} log(1 + x^3)/x^3 dx.
%F A186706 Equals Integral_{x=0..oo} log(1 + 4/(x*(x + 2))) dx. (End)
%e A186706 3.627598728468435701188156515284311464568132496185481151139769870776...
%t A186706 RealDigits[2 Pi/Sqrt[3], 10, 111][[1]] (* _Robert G. Wilson v_, Nov 18 2012 *)
%o A186706 (PARI) intnum(x=1e-7,1e6,eta(x*I,1)) \\ _Charles R Greathouse IV_, Feb 26 2011
%Y A186706 Cf. A073005, A073006, A093602, A093766,
%K A186706 cons,nonn
%O A186706 1,1
%A A186706 _Robert G. Wilson v_, Feb 25 2011