A221209 Decimal expansion of two times the Catalan constant.

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, 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, 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

Offset: 1

Author

R. J. Mathar, Feb 21 2013

Keywords

• nonn
• cons
• easy

Examples

			1.83193118835443803010920702986476822154...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.
• I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, 5th edition, Academic Press, 1994, eq. (3.521.2).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• E. D. Krupnikov and K. S. Kölbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95.
• Michael I. Shamos, A catalog of the real numbers, (2007). See p. 594.

Crossrefs

Cf. A006752, A247685.

Programs

Magma

SetDefaultRealField(RealField(100)); R:=RealField(); 2*Catalan(R); // G. C. Greubel, Aug 25 2018

Maple

evalf(2*Catalan) ;

Mathematica

RealDigits[2 Catalan, 10, 100][[1]] (* Bruno Berselli, Feb 21 2013 *)

PARI

default(realprecision, 100); 2*Catalan \\ G. C. Greubel, Aug 25 2018

Formula

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

Equals 2*A006752.

From Amiram Eldar, Aug 20 2020: (Start)

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

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

Equals 3F2(1/2,1,1;3/2,3/2;1) [Krupnikov]. - R. J. Mathar, May 13 2024

From Stefano Spezia, Nov 12 2024: (Start)

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).

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

Equals A247685/2. - Hugo Pfoertner, Nov 12 2024

Equals Sum_{n>=1} H(2*n) * binomial(2*n, n) / (4^n * (2*n + 1)), where H(n) is the n-th harmonic number. - Antonio Graciá Llorente, Apr 04 2025

Equals Integral_{x=-1..1} -log(abs(x))/(1 + x^2) dx. - Kritsada Moomuang, May 28 2025