This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A222362 #80 Feb 16 2025 08:33:19
%S A222362 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 A222362 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 A222362 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 A222362 Decimal expansion of the ratio of the area of the latus rectum segment of any equilateral hyperbola to the square of its semi-axis: sqrt(2) - log(1 + sqrt(2)).
%C A222362 Just as circles are ellipses whose semi-axes are equal (and are called the radius of the circle), equilateral (or rectangular) hyperbolas are hyperbolas whose semi-axes are equal.
%C A222362 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 semi-axis is the universal equilateral hyperbolic constant sqrt(2) - log(1 + sqrt(2)).
%C A222362 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 A222362 This constant is also the abscissa of the vertical asymptote of the involute of the logarithmic curve (starting point (1,0)). - _Jean-François Alcover_, Nov 25 2016
%D A222362 H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
%D A222362 P. Lockhart, Measurement, Harvard University Press, 2012, p. 369.
%H A222362 G. C. Greubel, <a href="/A222362/b222362.txt">Table of n, a(n) for n = 0..10000</a>
%H A222362 J.-F. Alcover, <a href="/A222362/a222362.pdf">Asymptote</a> of the logarithmic curve involute.
%H A222362 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 A222362 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Mathematical Constants, Errata and Addenda</a>, 2012, section 8.1.
%H A222362 J. Pahikkala, <a href="http://planetmath.org/arclengthofparabola">Arc Length Of Parabola</a>, PlanetMath.
%H A222362 Sylvester Reese and Jonathan Sondow, <a href="https://mathworld.wolfram.com/UniversalParabolicConstant.html">Universal Parabolic Constant</a>, MathWorld.
%H A222362 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RectangularHyperbola.html">Rectangular hyperbola</a>.
%H A222362 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hyperbola#Rectangular_hyperbola_with_horizontal.2Fvertical_asymptotes_.28Cartesian_coordinates.29">Equilateral hyperbola</a>.
%H A222362 Wikipedia, <a href="http://en.wikipedia.org/wiki/Universal_parabolic_constant">Universal parabolic constant</a>.
%H A222362 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A222362 Sqrt(2) - arcsinh(1), also equals Integral_{1..oo} 1/(x^2*(1+x)^(1/2)) dx. - _Jean-François Alcover_, Apr 16 2015
%F A222362 Equals Integral_{x = 0..1} x^2/sqrt(1 + x^2) dx. - _Peter Bala_, Feb 28 2019
%e A222362 0.532839975353552023569079399229905769541511547115312662423384129337355...
%p A222362 Digits:=100: evalf(sqrt(2)-arcsinh(1)); # _Wesley Ivan Hurt_, Nov 27 2016
%t A222362 RealDigits[Sqrt[2] - Log[1 + Sqrt[2]], 10, 111][[1]]
%o A222362 (PARI) sqrt(2)-log(sqrt(2)+1) \\ _Charles R Greathouse IV_, Apr 18 2013
%o A222362 (PARI) sqrt(2)-asinh(1) \\ _Charles R Greathouse IV_, Dec 04 2020
%o A222362 (Magma) Sqrt(2) - Log(Sqrt(2)+1); // _G. C. Greubel_, Feb 02 2018
%Y A222362 Cf. A002193, A091648, A103710, A103711, A180434, A278386.
%K A222362 cons,easy,nonn
%O A222362 0,1
%A A222362 Sylvester Reese and _Jonathan Sondow_, Mar 01 2013