A091506 Decimal expansion of (2 + sqrt(2) + 5*arcsinh(1))/9.

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

Offset

0,1

Comments

Average distance between two points chosen at random on two different edges of a unit square.

References

Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery. Natick: A K Peters (2004): p. 66, Example 57(a).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, Ten Problems in Experimental Mathematics

Eric Weisstein's World of Mathematics, Square Line Picking

Formula

Also equals (2 + sqrt(2) + 5*log(1 + sqrt(2)))/9. - Jean-François Alcover, Feb 14 2014

James D. Klein proved that this constant is equal to 2/3*int(int(sqrt(x^2 + y^2), x = 0..1), y = 0..1) + 1/3*int(int(sqrt(1 + (y - u)^2), u = 0..1), y = 0..1). - John M. Campbell, Apr 02 2016

Example

0.86900905527453446388497059434540662485671927963168056966035...

Mathematica

RealDigits[(2 + Sqrt[2] + 5ArcSinh[1])/9, 10, 120][[1]] (* Harvey P. Dale, May 22 2013 *)

Prog

(PARI) (2 + sqrt(2) + 5*asinh(1))/9 \\ G. C. Greubel, Aug 17 2018
(Magma) SetDefaultRealField(RealField(100)); (2 + Sqrt(2) + 5*Argsinh(1))/9; // G. C. Greubel, Aug 17 2018

Crossrefs

Cf. A091505.

Sequence in context: A110214 A305709 A093721 * A021539 A084893 A019939

Adjacent sequences: A091503 A091504 A091505 * A091507 A091508 A091509

Keyword

nonn,easy,cons

Author

Eric W. Weisstein, Jan 16 2004

Extensions

Broken link fixed by John M. Campbell, Apr 02 2016

Status

approved