A244921 Decimal expansion of (sqrt(2)+log(1+sqrt(2)))/3, the integral over the square [0,1]x[0,1] of sqrt(x^2+y^2) dx dy.

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

Offset: 0

Author

Jean-François Alcover, Jul 08 2014

Keywords

• nonn
• cons
• easy

Comments

This is also the expected distance from a randomly selected point in the unit square to a corner, as well as the expected distance from a randomly selected point in a 45-45-90 degree triangle of base length 1 to a vertex with an acute angle. - Derek Orr, Jul 27 2014

The average length of chords in a unit square drawn between two points uniformly and independently chosen at random on two adjacent sides. - Amiram Eldar, Aug 08 2020

Examples

			0.76519571646421269134476601639649679586594406787952782797665844888136988756...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• D. Bailey, J. Borwein, and R. Crandall, Advances in the theory of box integrals, Mathematics of Computation, Vol. 79, No. 271 (2010), pp. 1839-1866. See p. 1860.
• Philip W. Kuchel and Rodney J. Vaughan, Average lengths of chords in a square, Mathematics Magazine, Vol. 54, No. 5 (1981), pp. 261-269.
• Index entries for transcendental numbers

Crossrefs

Cf. A244920.

Programs

Mathematica

RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/3, 10, 104] // First

PARI

(sqrt(2)+log(1+sqrt(2)))/3 \\ G. C. Greubel, Jul 05 2017

Formula

Also equals (sqrt(2) + arcsinh(1))/3.

This is also 2*A103712. - Derek Orr, Jul 27 2014