A247674 Decimal expansion of the integral over the square [0,1]x[0,1] of sqrt(1+(x-y)^2) dx dy.

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

Offset: 1

Author

Jean-François Alcover, Sep 22 2014

Keywords

• nonn
• cons

Comments

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

Examples

			1.076635732895178008965379750243226282838269703135986...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• D. H. Bailey and J. M. Borwein, Highly Parallel, High-Precision Numerical Integration, Lawrence Berkeley National Laboratory (2005), p. 9.
• Philip W. Kuchel and Rodney J. Vaughan, Average lengths of chords in a square, Mathematics Magazine, Vol. 54, No. 5 (1981), pp. 261-269.

Crossrefs

Cf. A244921.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); (2 + Sqrt(2) + 5*Log(1+Sqrt(2)))/3; // G. C. Greubel, Aug 31 2018

Mathematica

RealDigits[2/3 - Sqrt[2]/3 + ArcSinh[1], 10, 103] // First

PARI

default(realprecision, 100); (2 + sqrt(2) + 5*log(1+sqrt(2)))/3 \\ G. C. Greubel, Aug 31 2018

Formula

Equals 2/3 - sqrt(2)/3 + arcsinh(1).

Equals 2*A244921 + A247674 = (2 + sqrt(2) + 5*log(1+sqrt(2)))/3.