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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. 8


%S 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,

%T 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,

%U 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

%N 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.

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

%C exp(d(2)) - exp(d(2))/Pi = 0.9994179247351742... ~ 1 - 1/1718. - _Gerald McGarvey_, Feb 21 2005

%C 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

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 8.1.

%D S. Reese, A universal parabolic constant, 2004, preprint.

%H Ivan Panchenko, <a href="/A103712/b103712.txt">Table of n, a(n) for n = 0..1000</a>

%H Steven. R. Finch, <a href="http://arxiv.org/abs/2001.00578">Mathematical Constants, Errata and Addenda</a>, 2012, section 8.1.

%H S. Reese, <a href="http://gaia.adelphi.edu/cgi-bin/makehtmlmov-css.pl?rtsp://gaia.adelphi.edu:554/General_Lectures/Pohle_Colloquiums/pohle200502.mov,pohle200502.mov,256,200">Pohle Colloquium Video Lecture: The universal parabolic constant, Feb 02 2005</a>

%H Sylvester Reese, Jonathan Sondow and Eric W. Weisstein, <a href="http://mathworld.wolfram.com/UniversalParabolicConstant.html">MathWorld: Universal Parabolic Constant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UniversalParabolicConstant.html">Universal Parabolic Constant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareLinePicking.html">Square Line Picking</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Universal_parabolic_constant">Universal parabolic constant</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

%F Equals Integral_{x>=1} arcsinh(x)/x^4 dx. - _Amiram Eldar_, Jun 26 2021

%e 0.38259785823210634567238300819824839793297203393976391398832922444...

%t RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/6, 10, 111][[1]] (* _Robert G. Wilson v_, Feb 14 2005 *)

%o (Maxima) fpprec: 100$ ev(bfloat((sqrt(2) + log(1 + sqrt(2)))/6)); /* _Martin Ettl_, Oct 17 2012 */

%o (PARI) (sqrt(2) + log(1 + sqrt(2)))/6 \\ _G. C. Greubel_, Sep 22 2017

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

%K cons,easy,nonn

%O 0,1

%A Sylvester Reese and _Jonathan Sondow_, Feb 13 2005

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Last modified July 28 18:41 EDT 2021. Contains 346335 sequences. (Running on oeis4.)