A346205 - OEIS

%I #13 Jun 12 2024 01:11:43

%S 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,

%T 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,

%U 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

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

%H G. C. Greubel, <a href="/A346205/b346205.txt">Table of n, a(n) for n = 0..10000</a>

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

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

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

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

%e 0.2288989481961786412366361253722...

%t x/.FindRoot[LambertW[-x]-LambertW[-1,-x]==2, {x, 0.1, 0.3}, WorkingPrecision -> 110]

%t RealDigits[2/(E^2-1)*Exp[2/(1-E^2)], 10, 135][[1]] (* _G. C. Greubel_, Jun 11 2024 *)

%o (PARI) exp(-cotanh(1))/sinh(1)

%o (Magma) SetDefaultRealField(RealField(135)); 2/(Exp(2)-1)*Exp(2/(1-Exp(2))); // _G. C. Greubel_, Jun 11 2024

%o (SageMath) numerical_approx(2/(e^2-1)*exp(2/(1-e^2)), digits=135) # _G. C. Greubel_, Jun 11 2024

%Y Cf. A072334, A073742, A073747.

%K nonn,cons

%O 0,1

%A _Gleb Koloskov_, Jul 10 2021