A091648 Decimal expansion of arccosh(sqrt(2)), the inflection point of sech(x).

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, 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, 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

Offset: 0

Author

Eric W. Weisstein, Jan 24 2004

Keywords

• nonn
• cons
• easy

Comments

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

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

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

Examples

			0.8813735870195430252326093249797923090281603282616...
		

References

• L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (85) page 16-17.
• E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, chapter 13, A Mapmaker's Paradise, pp. 163-180.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:4 at page 283.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• E. D. Krupnikov, K. S. Kölbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95.
• D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449.
• Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 614-615.
• R. Shrock and F. Y. Wu, Spanning trees on graphs and lattices in d dimensions, J Phys A: Math Gen 33 (2000) 3881-3902.
• Eric Weisstein's World of Mathematics, Hyperbolic Secant.
• Eric Weisstein's World of Mathematics, Universal Parabolic Constant.
• Index entries for transcendental numbers.

Crossrefs

Cf. A014176, A103710, A103711, A103712, A181048.

Programs

Mathematica

RealDigits[Log[1 + Sqrt[2]], 10, 100][[1]] (* Alonso del Arte, Aug 11 2011 *)

Maxima

fpprec : 100$ ev(bfloat(log(1 + sqrt(2)))); /* Martin Ettl, Oct 17 2012 */

PARI

asinh(1) \\ Michel Marcus, Oct 19 2014

Formula

Equals log(1 + sqrt(2)). - Jonathan Sondow, Mar 15 2005

Equals (1/2)*log(3+2*sqrt(2)) = A244920/2. - R. J. Mathar, May 14 2006

Equals Sum_{n>=1, n odd} binomial(2*n,n)/(n*4^n) [see Lehmer link]. - R. J. Mathar, Mar 04 2009

Equals arcsinh(1), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013

Equals asin(i)/i. - L. Edson Jeffery, Oct 19 2014

Equals (Pi/4) * 3F2(1/4, 1/2, 3/4; 1, 3/2; 1). - Jean-François Alcover, Apr 23 2015

Equals arctanh(sqrt(2)/2). - Amiram Eldar, Apr 22 2022

Equals lim_{n->oo} Sum_{k=1..n} 1/sqrt(n^2+k^2). - Amiram Eldar, May 19 2022

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

Equals 2F1(1/2,1/2;3/2;-1) [Krupnikov]. - R. J. Mathar, May 13 2024

Equals Integral_{x=0..1} (x^sqrt(2) - 1)/log(x) dx. - Kritsada Moomuang, Jun 06 2025