

A222362


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)).


4



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
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OFFSET

0,1


COMMENTS

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.
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)).
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)).  JeanFrançois 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


FORMULA

Sqrt(2)  arcsinh(1), also equals Integral_{1..infinity} 1/(x^2*(1+x)^(1/2)) dx.  JeanFrançois Alcover, Apr 16 2015


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
(MAGMA) Sqrt(2)  Log(Sqrt(2)+1) // G. C. Greubel, Feb 02 2018


CROSSREFS

Cf. A002193, A091648, A103710, A103711, A278386.
Sequence in context: A271523 A125844 A171025 * A176524 A268690 A065627
Adjacent sequences: A222359 A222360 A222361 * A222363 A222364 A222365


KEYWORD

cons,easy,nonn


AUTHOR

Sylvester Reese and Jonathan Sondow, Mar 01 2013


STATUS

approved



