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A085984 Decimal expansion of solution to e^x*(-1 + x) == (1 + x)/e^x. 9
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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 [A178225]. - 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

G. C. Greubel, Table of n, a(n) for n = 1..5000

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.

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.

EXAMPLE

1.1996786402577338339163698486411419442614587884186072...

MATHEMATICA

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

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.

Sequence in context: A157293 A146487 A195790 * A157245 A072908 A217695

Adjacent sequences:  A085981 A085982 A085983 * A085985 A085986 A085987

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Jul 06 2003

STATUS

approved

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Last modified April 8 14:52 EDT 2020. Contains 333314 sequences. (Running on oeis4.)