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A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

%I A085984 #107 Feb 16 2025 08:32:50
%S A085984 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 A085984 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 A085984 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 A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.
%C A085984 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 A085984 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 A085984 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 A085984 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 A085984 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 A085984 Steven R. Finch, Mathematical constants, Volume 94, Encyclopedia of mathematics and its applications, Cambridge University Press, 2003, p. 268.
%D A085984 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:9 at page 283.
%H A085984 G. C. Greubel, <a href="/A085984/b085984.txt">Table of n, a(n) for n = 1..5000</a>
%H A085984 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 A085984 Robert Ferréol, <a href="https://www.mathcurve.com/courbes2d.gb/tractrice/tractrice.shtml">Tractrix</a>, Mathcurve.
%H A085984 D. E. Knuth, <a href="https://cs.stanford.edu/~knuth/papers/whirlpool.pdf">Whirlpool Permutations</a>, May 05 2020.
%H A085984 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 A085984 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KeplersEquation.html">Kepler's Equation</a>.
%H A085984 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaplaceLimit.html">Laplace Limit</a>.
%H A085984 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HyperbolicCotangent.html">Hyperbolic Cotangent</a>
%H A085984 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tractrix">Tractrix</a>.
%H A085984 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A085984 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 A085984 1.1996786402577338339163698486411419442614587884186072...
%t A085984 RealDigits[ x /. FindRoot[ Coth[x] == x, {x, 1}, WorkingPrecision -> 102]] // First (* _Jean-François Alcover_, Feb 08 2013 *)
%t A085984 1+2 NSum[LaguerreL[n-1,1,4 n]/n Exp[-2 n],{n,1,Infinity}] //
%t A085984   (* _Aaron Hendrickson_, Mar 17 2021 *)
%o A085984 (PARI) solve(u=1,2,tanh(u)-1/u)  /* type e.g. \p99 to get 99 digits; _M. F. Hasler_, Feb 01 2011 */
%Y A085984 Cf. A003957 (x = cos(x)), A009379, A033259, A069855, A209289.
%K A085984 nonn,cons
%O A085984 1,3
%A A085984 _Eric W. Weisstein_, Jul 06 2003