%I
%S 5,3,2,8,3,9,9,7,5,3,5,3,5,5,2,0,2,3,5,6,9,0,7,9,3,9,9,2,2,9,9,0,5,7,
%T 6,9,5,4,1,5,1,1,5,4,7,1,1,5,3,1,2,6,6,2,4,2,3,3,8,4,1,2,9,3,3,7,3,5,
%U 5,2,9,4,2,4,0,0,8,0,9,5,1,0,1,6,6,8,0,6,4,2,4,1,7,3,8,5,5,2,9,8,7,8,2,7,4,0,3,0,0,3
%N Decimal expansion of the ratio of the area of the latus rectum segment of any equilateral hyperbola to the square of its semiaxis: sqrt(2)  log(1 + sqrt(2)).
%C Just as circles are ellipses whose semiaxes are equal (and are called the radius of the circle), equilateral (or rectangular) hyperbolas are hyperbolas whose semiaxes are equal.
%C Just as the ratio of the area of a circle to the square of its radius is always Pi, the ratio of the area of the latus rectum segment of any equilateral hyperbola to the square of its semiaxis is the universal equilateral hyperbolic constant sqrt(2)  log(1 + sqrt(2)).
%C Note the remarkable similarity to sqrt(2) + log(1 + sqrt(2)), the universal parabolic constant A103710, which is a ratio of arc lengths rather than of areas. Lockhart (2012) says "the arc length integral for the parabola ... is intimately connected to the hyperbolic area integral ... I think it is surprising and wonderful that the length of one conic section is related to the area of another".
%C This constant is also the abscissa of the vertical asymptote of the involute of the logarithmic curve (starting point (1,0)).  _JeanFrançois Alcover_, Nov 25 2016
%D H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
%D P. Lockhart, Measurement, Harvard University Press, 2012, p. 369.
%H G. C. Greubel, <a href="/A222362/b222362.txt">Table of n, a(n) for n = 0..10000</a>
%H J.F. Alcover, <a href="/A222362/a222362.pdf">Asymptote</a> of the logarithmic curve involute.
%H I.N. Bronshtein, <a href="http://books.google.com/books?id=gCgOoMpluh8C&lpg=PA202&vq=%22areas%20in%20the%20hyperbola%22&pg=PA202#v=onepage&q&f=false">Handbook of Mathematics</a>, 5th ed., Springer, 2007, p. 202, eq. (3.338a).
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/erradd.pdf">Mathematical Constants, Errata and Addenda</a>, 2012, section 8.1.
%H J. Pahikkala, <a href="http://planetmath.org/arclengthofparabola">Arc Length Of Parabola</a>, PlanetMath.
%H S. Reese, J. Sondow, <a href="http://mathworld.wolfram.com/UniversalParabolicConstant.html">Universal Parabolic Constant</a>, MathWorld
%H E.W. Weisstein, <a href="http://mathworld.wolfram.com/RectangularHyperbola.html">Rectangular hyperbola</a>, MathWorld
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hyperbola#Rectangular_hyperbola_with_horizontal.2Fvertical_asymptotes_.28Cartesian_coordinates.29">Equilateral hyperbola</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Universal_parabolic_constant">Universal parabolic constant</a>
%F Sqrt(2)  arcsinh(1), also equals Integral_{1..infinity} 1/(x^2*(1+x)^(1/2)) dx.  _JeanFrançois Alcover_, Apr 16 2015
%e 0.532839975353552023569079399229905769541511547115312662423384129337355...
%p Digits:=100: evalf(sqrt(2)arcsinh(1)); # _Wesley Ivan Hurt_, Nov 27 2016
%t RealDigits[Sqrt[2]  Log[1 + Sqrt[2]], 10, 111][[1]]
%o (PARI) sqrt(2)log(sqrt(2)+1) \\ _Charles R Greathouse IV_, Apr 18 2013
%o (MAGMA) Sqrt(2)  Log(Sqrt(2)+1) // _G. C. Greubel_, Feb 02 2018
%Y Cf. A002193, A091648, A103710, A103711, A278386.
%K cons,easy,nonn
%O 0,1
%A Sylvester Reese and _Jonathan Sondow_, Mar 01 2013
