A085984 - OEIS

%I #99 Dec 19 2024 07:26:58

%S 1,1,9,9,6,7,8,6,4,0,2,5,7,7,3,3,8,3,3,9,1,6,3,6,9,8,4,8,6,4,1,1,4,1,

%T 9,4,4,2,6,1,4,5,8,7,8,8,4,1,8,6,0,7,2,0,8,9,1,5,4,7,7,7,8,3,9,1,8,1,

%U 2,4,7,2,5,2,2,3,8,4,7,4,7,9,9,9,9,0,8,6,9,9,2,1,4,6,5,0,9,3,7,9,8,8

%N Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

%C This constant can also be defined as the root of coth x = x, as this equation and the above are equivalent. - _Carl R. White_, Dec 09 2003. Also the root of x*tanh x = 1. - _N. J. A. Sloane_, May 07 2020

%C This constant is also the point on the parametric tractrix (t - tanh(t), sech(t)) the least distant from the origin. - _Michael Clausen_, Feb 18 2013

%C This constant also equals sqrt(lambda^2+1), where lambda is the Laplace limit constant A033259. - _Jean-François Alcover_, Sep 08 2014, after Steven Finch.

%C For each of the real symmetric n X n matrices M defined by M(i,j) = max(i,j) with n >= 2, there exist n-1 negative eigenvalues < -1/4 and only one positive eigenvalue lambda(n) such that n^2/2 < lambda(n) < n^2. Indeed, when n tends to infinity, lambda(n) ~ n^2/(this constant)^2 (see reference O. Carton et al.). For n = 2, the positive eigenvalue is (3+sqrt(17))/2 [A178255]. - _Bernard Schott_, Mar 13 2020

%D O. Carton, L. Rosaz, M. Zeitoun, Problèmes corrigés de Mathématiques posés au Concours de Mines/Ponts, Tome 5, Ellipses, 1992; Problème Mines-Ponts 1991 - Options M, P', TA - Epreuve pratique p. 125.

%D Steven R. Finch, Mathematical constants, Volume 94, Encyclopedia of mathematics and its applications, Cambridge University Press, 2003, p. 268.

%H G. C. Greubel, <a href="/A085984/b085984.txt">Table of n, a(n) for n = 1..5000</a>

%H Jogundas Armaitis, <a href="https://web.science.uu.nl/ITF/Teaching/2011/JogundasArmaitis.pdf">Molecules and Polarons in Extremely Imbalanced Fermi Mixtures</a>, Master's Thesis, Aug 11 2011, Institute for Theoretical Physics, Utrecht University.

%H Robert Ferréol, <a href="https://www.mathcurve.com/courbes2d.gb/tractrice/tractrice.shtml">Tractrix</a>, Mathcurve.

%H D. E. Knuth, <a href="https://cs.stanford.edu/~knuth/papers/whirlpool.pdf">Whirlpool Permutations</a>, May 05 2020.

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4393448/is-there-a-closed-form-of-the-laplace-limit-constant-x-such-that-fracxe#4401425">Laplace limit constant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KeplersEquation.html">Kepler's Equation</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaplaceLimit.html">Laplace Limit</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicCotangent.html">Hyperbolic Cotangent</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tractrix">Tractrix</a>.

%F Equals 1 + 2*Sum_{n>=1} (Laguerre(n-1,1,4n)/n)*e^(-2n) (see Mathematics Stack Exchange in Links). - _Aaron Hendrickson_, Mar 17 2022

%e 1.1996786402577338339163698486411419442614587884186072...

%t RealDigits[ x /. FindRoot[ Coth[x] == x, {x, 1}, WorkingPrecision -> 102]] // First (* _Jean-François Alcover_, Feb 08 2013 *)

%t 1+2 NSum[LaguerreL[n-1,1,4 n]/n Exp[-2 n],{n,1,Infinity}] //

%t   (* _Aaron Hendrickson_, Mar 17 2021 *)

%o (PARI) solve(u=1,2,tanh(u)-1/u)  /* type e.g. \p99 to get 99 digits; _M. F. Hasler_, Feb 01 2011 */

%Y Cf. A003957 (x = cos(x)), A009379, A033259, A069855, A209289.

%K nonn,cons,changed

%O 1,3

%A _Eric W. Weisstein_, Jul 06 2003