|
| |
|
|
A091648
|
|
Decimal expansion of arccosh(sqrt(2)), the inflection point of sech(x).
|
|
24
|
|
|
|
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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
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
|
|
|
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.
|
|
|
LINKS
|
Ivan Panchenko, Table of n, a(n) for n = 0..1000
D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449.
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.
|
|
|
FORMULA
|
Equals log(1 + sqrt(2)). - Jonathan Sondow, Mar 15 2005
Equals (1/2)*log(3+2*sqrt(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
|
|
|
EXAMPLE
|
0.8813735870195430252326093249797923090281603282616...
|
|
|
MATHEMATICA
|
RealDigits[Log[1 + Sqrt[2]], 10, 100][[1]] (* Alonso del Arte, Aug 11 2011 *)
|
|
|
PROG
|
(Maxima) fpprec : 100$ ev(bfloat(log(1 + sqrt(2)))); /* Martin Ettl, Oct 17 2012 */
(PARI) asinh(1) \\ Michel Marcus, Oct 19 2014
|
|
|
CROSSREFS
|
Cf. A103710, A103711, A103712, A181048.
Sequence in context: A176155 A174127 A230153 * A135707 A021923 A296496
Adjacent sequences: A091645 A091646 A091647 * A091649 A091650 A091651
|
|
|
KEYWORD
|
nonn,cons,easy
|
|
|
AUTHOR
|
Eric W. Weisstein, Jan 24 2004
|
|
|
STATUS
|
approved
|
| |
|
|
