A103712 Decimal expansion of the expected distance from a randomly selected point in the unit square to its center: (sqrt(2) + log(1 + sqrt(2)))/6.

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

Offset: 0

Author

Sylvester Reese and Jonathan Sondow, Feb 13 2005

Keywords

• cons
• easy
• nonn

Comments

Is it a coincidence that this constant is equal to 1/6 of the universal parabolic constant A103710? (Reese, 2004; Finch, 2012)

exp(d(2)) - exp(d(2))/Pi = 0.9994179247351742... ~ 1 - 1/1718. - Gerald McGarvey, Feb 21 2005

Take a point on a line of irrational slope and a line segment of a given length centered at the point, integrate the distance of a point on the line to the set of lattice points along the line segment, and divide by the length. The limit as the length approaches infinity can be shown by a generalization of the Equidistribution Theorem to give the expected distance of a point in the unit square to its corners, this constant. - Thomas Anton, Jun 19 2021

Examples

			0.38259785823210634567238300819824839793297203393976391398832922444...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 8.1.
• S. Reese, A universal parabolic constant, 2004, preprint.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Steven R. Finch, Mathematical Constants, Errata and Addenda, 2012, section 8.1.
• Sylvester Reese, Pohle Colloquium Video Lecture: The universal parabolic constant, February 2005.
• Sylvester Reese, Jonathan Sondow and Eric W. Weisstein, MathWorld: Universal Parabolic Constant.
• Eric Weisstein's World of Mathematics, Universal Parabolic Constant.
• Eric Weisstein's World of Mathematics, Square Line Picking.
• Wikipedia, Universal parabolic constant.
• Index entries for transcendental numbers.

Crossrefs

Equal to (A002193 + A091648)/6 = (A103710)/6 = (A103711)/3.

Cf. A244921.

Programs

Mathematica

RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/6, 10, 111][[1]] (* Robert G. Wilson v, Feb 14 2005 *)

Maxima

fpprec: 100$ ev(bfloat((sqrt(2) + log(1 + sqrt(2)))/6)); /* Martin Ettl, Oct 17 2012 */

PARI

(sqrt(2) + log(1 + sqrt(2)))/6 \\ G. C. Greubel, Sep 22 2017

Formula

Equals (1/3)*Integral_{x = 0..1} sqrt(1 + x^2) dx. - Peter Bala, Feb 28 2019

Equals Integral_{x>=1} arcsinh(x)/x^4 dx. - Amiram Eldar, Jun 26 2021

Equals A244921 / 2. - Amiram Eldar, Jun 04 2023