|
%I #87 Mar 21 2024 06:44:01
%S 8,8,1,3,7,3,5,8,7,0,1,9,5,4,3,0,2,5,2,3,2,6,0,9,3,2,4,9,7,9,7,9,2,3,
%T 0,9,0,2,8,1,6,0,3,2,8,2,6,1,6,3,5,4,1,0,7,5,3,2,9,5,6,0,8,6,5,3,3,7,
%U 7,1,8,4,2,2,2,0,2,6,0,8,7,8,3,3,7,0,6,8,9,1,9,1,0,2,5,6,0,4,2,8,5,6
%N Decimal expansion of arccosh(sqrt(2)), the inflection point of sech(x).
%C Asymptotic growth constant in the exponent for the number of spanning trees on the 2 X infinity strip on the square lattice. - _R. J. Mathar_, May 14 2006
%C Arccosh(sqrt(2)) = (1/2)*log((sqrt(2)+1)/(sqrt(2)-1)) = log(tan(3*Pi/8)) = int(1/cos(x),x=0..Pi/4). Therefore, in Gerardus Mercator's (conformal) map this is the value of the ordinate y/R (R radius of the spherical earth) for latitude phi = 45 degrees north, or Pi/4. See, e.g., the Eli Maor reference, eqs. (5) and (6). This is the latitude of, e.g., the Mission Point Lighthouse, Michigan, U.S.A. - _Wolfdieter Lang_, Mar 05 2013
%C Decimal expansion of the arclength on the hyperbola y^2 - x^2 = 1 from (0,0) to (1,sqrt(2)). - _Clark Kimberling_, Jul 04 2020
%D L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (85) page 16-17.
%D E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, chapter 13, A Mapmaker's Paradise, pp. 163-180.
%H Ivan Panchenko, <a href="/A091648/b091648.txt">Table of n, a(n) for n = 0..1000</a>
%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting Series Involving the Central Binomial Coefficient</a>, Am. Math. Monthly 92 (1985) 449.
%H R. Shrock and F. Y. Wu, <a href="http://dx.doi.org/10.1088/0305-4470/33/21/303">Spanning trees on graphs and lattices in d dimensions</a>, J Phys A: Math Gen 33 (2000) 3881-3902.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicSecant.html">Hyperbolic Secant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UniversalParabolicConstant.html">Universal Parabolic Constant</a>.
%F Equals log(1 + sqrt(2)). - _Jonathan Sondow_, Mar 15 2005
%F Equals (1/2)*log(3+2*sqrt(2)). - _R. J. Mathar_, May 14 2006
%F Equals Sum_{n>=1, n odd} binomial(2*n,n)/(n*4^n) [see Lehmer link]. - _R. J. Mathar_, Mar 04 2009
%F Equals arcsinh(1), since arcsinh(x) = log(x+sqrt(x^2+1)). - _Stanislav Sykora_, Nov 01 2013
%F Equals asin(i)/i. - _L. Edson Jeffery_, Oct 19 2014
%F Equals (Pi/4) * 3F2(1/4, 1/2, 3/4; 1, 3/2; 1). - _Jean-François Alcover_, Apr 23 2015
%F Equals arctanh(sqrt(2)/2). - _Amiram Eldar_, Apr 22 2022
%F Equals lim_{n->oo} Sum_{k=1..n} 1/sqrt(n^2+k^2). - _Amiram Eldar_, May 19 2022
%F Equals Sum_{n >= 1} 1/(n*P(n, sqrt(2))*P(n-1, sqrt(2))), where P(n, x) denotes the n-th Legendre polynomial. The first twenty terms of the series gives the approximate value 0.88137358701954(24...), correct to 14 decimal places. - _Peter Bala_, Mar 16 2024
%e 0.8813735870195430252326093249797923090281603282616...
%t RealDigits[Log[1 + Sqrt[2]], 10, 100][[1]] (* _Alonso del Arte_, Aug 11 2011 *)
%o (Maxima) fpprec : 100$ ev(bfloat(log(1 + sqrt(2)))); /* _Martin Ettl_, Oct 17 2012 */
%o (PARI) asinh(1) \\ _Michel Marcus_, Oct 19 2014
%Y Cf. A014176, A103710, A103711, A103712, A181048.
%K nonn,cons,easy
%O 0,1
%A _Eric W. Weisstein_, Jan 24 2004
|
