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

1,2

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.

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

Example

1.83193118835443803010920702986476822154...

Maple

evalf(2*Catalan) ;

Mathematica

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

Prog

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

Crossrefs

Cf. A006752, A247685.

Sequence in context: A097890 A088453 A019782 * A056030 A364889 A195473

Adjacent sequences: A221206 A221207 A221208 * A221210 A221211 A221212

Keyword

nonn,cons,easy

Author

R. J. Mathar, Feb 21 2013

Status

approved