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A085984
Decimal expansion of solution to e^x*(-1 + x) == (1 + x)/e^x.
13
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, 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, 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
OFFSET
1,3
COMMENTS
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
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
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.
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
REFERENCES
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.
Steven R. Finch, Mathematical constants, Volume 94, Encyclopedia of mathematics and its applications, Cambridge University Press, 2003, p. 268.
LINKS
Jogundas Armaitis, Molecules and Polarons in Extremely Imbalanced Fermi Mixtures, Master's Thesis, Aug 11 2011, Institute for Theoretical Physics, Utrecht University.
Robert Ferréol, Tractrix, Mathcurve.
D. E. Knuth, Whirlpool Permutations, May 05 2020.
Mathematics Stack Exchange, Laplace limit constant.
Eric Weisstein's World of Mathematics, Kepler's Equation
Eric Weisstein's World of Mathematics, Laplace Limit
Eric Weisstein's World of Mathematics, Hyperbolic Cotangent
Wikipedia, Tractrix.
FORMULA
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
EXAMPLE
1.1996786402577338339163698486411419442614587884186072...
MATHEMATICA
RealDigits[ x /. FindRoot[ Coth[x] == x, {x, 1}, WorkingPrecision -> 102]] // First (* Jean-François Alcover, Feb 08 2013 *)
1+2 NSum[LaguerreL[n-1, 1, 4 n]/n Exp[-2 n], {n, 1, Infinity}] //
(* Aaron Hendrickson, Mar 17 2021 *)
PROG
(PARI) solve(u=1, 2, tanh(u)-1/u) /* type e.g. \p99 to get 99 digits; M. F. Hasler, Feb 01 2011 */
CROSSREFS
Cf. A003957 (x = cos(x)), A009379, A033259, A069855, A209289.
Sequence in context: A346442 A195790 A346444 * A335011 A157245 A334846
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jul 06 2003
STATUS
approved