A091505 Decimal expansion of (2 + sqrt(2) + 5*arcsinh(1))/15.

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

Offset: 0

Author

Eric W. Weisstein, Jan 16 2004

Keywords

• nonn
• cons

Comments

Or equally, decimal expansion of (2 + sqrt(2) + 5*log(1+sqrt(2)))/15.

Average distance between two points chosen at random in a unit square.

Examples

			0.5214054331647206783309823566...
		

References

• S. R. Finch, Mathematical Constants, Cambridge, 2003, Sections 8.1 p. 479 and 8.5 p.498.
• L. A. Santalo, Integral Geometry and Geometric Probability, Addison-Wesley, 1976, see p. 49.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Uwe Bäsel, The moments of the distance between two random points in a regular polygon, arXiv:2101.03815 [math.PR], 2021.
• Jens Egholm Pedersen, Jörg Conradt, and Tony Lindeberg, Covariant spatio-temporal receptive fields for neuromorphic computing, arXiv:2405.00318 [cs.NE], 2024. See p. 12.
• Michael Penn, The average distance between points on a square, video (2022).
• Eric Weisstein's World of Mathematics, Square Line Picking
• Index entries for transcendental numbers

Crossrefs

Cf. A091506, A073012.

Programs

Mathematica

RealDigits[(2+Sqrt[2]+5ArcSinh[1])/15,10,120][[1]] (* Harvey P. Dale, Jul 18 2011 *)

PARI

(2 + sqrt(2) + 5*asinh(1))/15 \\ G. C. Greubel, Jan 11 2017

PARI

(2 + sqrt(2) + 5*log(sqrt(2)+1))/15 \\ Charles R Greathouse IV, Nov 21 2024