A221209 - OEIS

%I #36 Nov 12 2024 17:23:37

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

%T 2,2,1,5,4,8,2,9,8,7,4,8,5,6,3,3,4,4,2,6,8,5,3,2,9,9,6,2,3,9,2,4,3,5,

%U 2,6,0,3,9,5,5,2,5,0,9,5,3,8,9,5,8,7,1,3,0,2,5,8,5,2,2,3,0,2,1,2

%N Decimal expansion of two times the Catalan constant.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

%D I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, 5th edition, Academic Press, 1994, eq. (3.521.2).

%H G. C. Greubel, <a href="/A221209/b221209.txt">Table of n, a(n) for n = 1..10000</a>

%H E. D. Krupnikov and K. S. Kölbig, <a href="https://doi.org/10.1016/S0377-0427(96)00111-2">Some special cases of the generalized hypergeometric function (q+1)Fq</a>, J. Comp. Appl. Math. 78 (1997) 79-95.

%H 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. 594.

%F Equals Integral_{x=0..oo} x/cosh(x) dx.

%F Equals 2*A006752.

%F From _Amiram Eldar_, Aug 20 2020: (Start)

%F Equals Integral_{x=0..Pi/2} x/sin(x) dx.

%F Equals 1 + Integral_{x=0..oo} x * exp(-x) * tanh(x) dx. (End)

%F Equals 3F2(1/2,1,1;3/2,3/2;1) [Krupnikov]. - _R. J. Mathar_, May 13 2024

%F From _Stefano Spezia_, Nov 12 2024: (Start)

%F Equals Integral_{x=0..oo} arctan(x)/(x*sqrt(x^2 + 1)) dx = Integral_{x=0..1} K(x^2) dx, where K(x) is the complete elliptic integral of the first kind (see Shamos).

%F Equals Sum_{k>=0} 2^(2*k)/((2*k + 1)^2*binomial(2*k,k)) (see Finch). (End)

%F Equals A247685/2. - _Hugo Pfoertner_, Nov 12 2024

%e 1.83193118835443803010920702986476822154...

%p evalf(2*Catalan) ;

%t RealDigits[2 Catalan, 10, 100][[1]] (* _Bruno Berselli_, Feb 21 2013 *)

%o (PARI) default(realprecision, 100); 2*Catalan \\ _G. C. Greubel_, Aug 25 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); 2*Catalan(R); // _G. C. Greubel_, Aug 25 2018

%Y Cf. A006752, A247685.

%K nonn,cons,easy

%O 1,2

%A _R. J. Mathar_, Feb 21 2013