A346205 Decimal expansion of solution to LambertW(-x) - LambertW(-1,-x) = 2.

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

Offset: 0

Author

Gleb Koloskov, Jul 10 2021

Keywords

• nonn
• cons

Examples

			0.2288989481961786412366361253722...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A072334, A073742, A073747.

Programs

Magma

SetDefaultRealField(RealField(135)); 2/(Exp(2)-1)*Exp(2/(1-Exp(2))); // G. C. Greubel, Jun 11 2024

Mathematica

x/.FindRoot[LambertW[-x]-LambertW[-1,-x]==2, {x, 0.1, 0.3}, WorkingPrecision -> 110]
RealDigits[2/(E^2-1)*Exp[2/(1-E^2)], 10, 135][[1]] (* G. C. Greubel, Jun 11 2024 *)

PARI

exp(-cotanh(1))/sinh(1)

SageMath

numerical_approx(2/(e^2-1)*exp(2/(1-e^2)), digits=135) # G. C. Greubel, Jun 11 2024

Formula

Equals exp(-coth(1))/sinh(1) = exp(-A073747)/A073742.

Equals (coth(1)-1)*exp(1-coth(1)) = (A073747-1)*exp(1-A073747).

Equals (coth(1)+1)/exp(1+coth(1)) = (A073747+1)/exp(1+A073747).

Equals 2/(e^2-1)*exp(2/(1-e^2)) = 2/(A072334^2-1)*exp(2/(1-A072334^2)).