OFFSET
0,1
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
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.
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
EXAMPLE
0.38259785823210634567238300819824839793297203393976391398832922444...
MATHEMATICA
RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/6, 10, 111][[1]] (* Robert G. Wilson v, Feb 14 2005 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Sylvester Reese and Jonathan Sondow, Feb 13 2005
STATUS
approved