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 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)). 5
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. 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)). 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". 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 REFERENCES H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58. P. Lockhart, Measurement, Harvard University Press, 2012, p. 369. LINKS G. C. Greubel, Table of n, a(n) for n = 0..10000 J.-F. Alcover, Asymptote of the logarithmic curve involute. I.N. Bronshtein, Handbook of Mathematics, 5th ed., Springer, 2007, p. 202, eq. (3.338a). S. R. Finch, Mathematical Constants, Errata and Addenda, 2012, section 8.1. J. Pahikkala, Arc Length Of Parabola, PlanetMath. S. Reese, J. Sondow, Universal Parabolic Constant, MathWorld E.W. Weisstein, Rectangular hyperbola, MathWorld Wikipedia, Equilateral hyperbola Wikipedia, Universal parabolic constant Index entries for transcendental numbers FORMULA Sqrt(2) - arcsinh(1), also equals Integral_{1..infinity} 1/(x^2*(1+x)^(1/2)) dx. - Jean-François Alcover, Apr 16 2015 Equals Integral_{x = 0..1} x^2/sqrt(1 + x^2) dx. - Peter Bala, Feb 28 2019 EXAMPLE 0.532839975353552023569079399229905769541511547115312662423384129337355... MAPLE Digits:=100: evalf(sqrt(2)-arcsinh(1)); # Wesley Ivan Hurt, Nov 27 2016 MATHEMATICA RealDigits[Sqrt[2] - Log[1 + Sqrt[2]], 10, 111][[1]] PROG (PARI) sqrt(2)-log(sqrt(2)+1) \\ Charles R Greathouse IV, Apr 18 2013 (PARI) sqrt(2)-asinh(1) \\ Charles R Greathouse IV, Dec 04 2020 (Magma) Sqrt(2) - Log(Sqrt(2)+1) // G. C. Greubel, Feb 02 2018 CROSSREFS Cf. A002193, A091648, A103710, A103711, A180434, A278386. Sequence in context: A352903 A125844 A171025 * A352024 A176524 A309276 Adjacent sequences: A222359 A222360 A222361 * A222363 A222364 A222365 KEYWORD cons,easy,nonn AUTHOR Sylvester Reese and Jonathan Sondow, Mar 01 2013 STATUS approved

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