A103712 - OEIS

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

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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.

Index entries for transcendental numbers.