A069855 -id:A069855 - cp's OEIS frontend

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

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

Eric W. Weisstein, Jul 06 2003

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

			1.1996786402577338339163698486411419442614587884186072...

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.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:9 at page 283.

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.
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.
Index entries for transcendental numbers.

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

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 *)

solve(u=1,2,tanh(u)-1/u)  /* type e.g. \p99 to get 99 digits; M. F. Hasler, Feb 01 2011 */

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

A009399 Expansion of log(1+tanh(x)*x).

Original entry on oeis.org

0, 2, -20, 576, -33312, 3258880, -485139456, 102300807168, -29028932390912, 10668077137133568, -4929291212351078400, 2797060130323340197888, -1912137417504544127975424, 1550018044651811766917922816

Offset: 0

R. H. Hardin

sign

Cf. A009379, A069855, A085984.

With[{nn=30},Take[CoefficientList[Series[Log[1+Tanh[x]*x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jun 13 2016 *)

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = A069855 = 0.8603335890193797624838934241376623334118843632... is the root of the equation r * tan(r) = 1. - Vaclav Kotesovec, Dec 21 2017

Extended with signs by Olivier Gérard, Mar 15 1997

Prior Mathematica program replaced by Harvey P. Dale, Jun 13 2016

A346062 Decimal expansion of the minimum value of the area of a rhombus circumscribed around a cosine-shaped lens, whose vertices lie on coordinate axes.

Original entry on oeis.org

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

Offset: 1

Gleb Koloskov, Jul 03 2021

Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and a rhombus circumscribed around S, whose vertices lie on coordinate axes.
This constant represents the value of the minimum area of such a rhombus KLMN with vertices K(0,2v), L(-2u,0), M(0,-2v), N(2u,0).
The rhombus touches S at the midpoints of its sides, A(u,v), B(-u,v), C(-u,-v), D(u,-v) which define a rectangle ABCD of the maximum area, inscribed in S, whose sides are parallel to coordinate axes. The constant u can be found as a root of equation x=cot(x) and is known as A069855, and v=cos(u)=u/sqrt(1+u^2).

			4.4887707055283605403232300252898136708822792436449257365...

Gleb Koloskov, Geometric illustration

Cf. A069855.

N[Minimize[{2 (x+Cot[x])^2 Sin[x],{x>0,x

u=solve(x=0.5,1,x-cotan(x));8*u^2/sqrt(1+u^2)

Equals 8*A069855^2/sqrt(1+A069855^2).

A153741 Number of elements in wreath product C_2 wr S_n that alternate up/not-up with respect to a weak product ordering.

Original entry on oeis.org

2, 3, 14, 49, 376, 1987, 21328, 150337, 2074624, 18279971, 308317184, 3259985969, 64981320704, 801591982115, 18436312819712, 259914703640065, 6774998673915904, 107452993132016323, 3130412454801965056

Offset: 1

Andrew Niedermaier, Dec 31 2008

nonn

			Viewing elements in one-line notation as a list of ordered pairs with first entries in [2] and second entries forming a permutation in S_n, two of the 6 up/not-up elements for n=3 are (1,2) (2,3) (1,1) and (1,1) (1,3) (2,2). Note that the first element goes up/down and the second goes up/not-up with respect to the weak product ordering on ordered pairs.

G. C. Greubel, Table of n, a(n) for n = 1..400
A. Niedermaier and J. Remmel, Analogues of Up-down Permutations for Colored Permutations, J. Int. Seq. 13 (2010), 10.5.6.

Cf. A069855.

Rest[CoefficientList[Series[(1+Sin[x]+x*Cos[x])/(Cos[x]-x*Sin[x]), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 25 2013 *)

E.g.f.: (1 + sin(x) + x*cos(x))/(cos(x) - x*sin(x)).

a(n) ~ c * n! / r^(n+1), where r = 0.860333589... (=A069855) is the root of the equation sin(r)*r = cos(r), and c = 2/((2+r^2)*sin(r)) = 0.9628268573779... if n is even and c = 2-2/(r^2+2*r*tan(r)) = 1.2701193119933... if n is odd. - Vaclav Kotesovec, Sep 25 2013

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

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A009399 Expansion of log(1+tanh(x)*x).

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A346062 Decimal expansion of the minimum value of the area of a rhombus circumscribed around a cosine-shaped lens, whose vertices lie on coordinate axes.

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A153741 Number of elements in wreath product C_2 wr S_n that alternate up/not-up with respect to a weak product ordering.

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