A091505 - OEIS

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

This average was apparently first calculated by the Austrian mathematician Emanuel Czuber (1851-1925) in 1884. - Amiram Eldar, Mar 26 2026

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Sections 8.1, p. 479 and 8.5, p.498.

Luis A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, 1976, see p. 49.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

Narine G. Aharonyan and Victor K. Ohanyan, Moments of the distance between two random points, Modeling of Artificial Intelligence, Vol. 10, No. 2 (2016), pp. 64-70. See p. 67, eq. (3.2).

Uwe Bäsel, The moments of the distance between two random points in a regular polygon, arXiv:2101.03815 [math.PR], 2021.

Ernesto Bonomi and Jean-Luc Lutton, The N-City Travelling Salesman Problem: Statistical Mechanics and the Metropolis Algorithm, SIAM Review, Vol. 26, No. 4 (1984), pp. 551-568; ResearchGate link. See p. 559, eq. (29).

Emanuel Czuber, Geometrische Wahrscheinlichkeiten und Mittelwerte, Leipzig, B. G. Teubner, 1884, p. 204.

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.

FORMULA

Equals (2 + sqrt(2) + 5*log(1+sqrt(2)))/15.

EXAMPLE

0.521405433164720678330982356607243974914031567779008...

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A091506, A073012.

Cf. this constant (square), A093064 (triangle), A093070 (disk), A394596 (pentagon), A394597 (hexagon), A394598 (octagon), A394599 (10-gon), A394600 (12-gon), A394601 (rhombus), A394602 (rectangle).

Sequence in context: A348186 A356030 A266455 * A366626 A030357 A212879

Adjacent sequences: A091502 A091503 A091504 * A091506 A091507 A091508

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Jan 16 2004

STATUS

approved