A377635 - OEIS

%I #30 Nov 21 2024 17:55:58

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

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

%U 2,1,3,5,2,3,0,9,3,7,1,9,1,3,3,7,3,3,9,6,2,0

%N Decimal expansion of 1/(exp(2) - 1).

%H Michael Penn, <a href="https://www.youtube.com/watch?v=9_Klka9x4zg">One of my favorite identities</a>, YouTube video, 2023.

%H Michael Ian Shamos, <a href="https://citeseerx.ist.psu.edu/pdf/ae33a269baba5e8b1038e719fb3209e8a00abec5">Shamos's catalog of the real numbers</a>, 2011, p. 218.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%H <a href="/index/Z#zeta_function">Index entries for sequences related to zeta function</a>.

%F Equals 1/(A072334 - 1).

%F Equals Sum_{k >= 1} (-1)^(k+1)*zeta(2*k)/Pi^(2*k).

%F From _Amiram Eldar_, Nov 08 2024: (Start)

%F Formulas from Shamos (2011):

%F Equals (coth(1) - 1)/2 = (A073747 - 1)/2.

%F Equals Sum_{k>=1} exp(-2*k).

%F Equals Sum_{k>=1} 1/(k^2*Pi^2 + 1).

%F Equals Sum_{k>=0} B(k)*2^(k-1)/k!, where B(k) = A027641(k)/A027642(k) is the k-th Bernoulli number. (End)

%e 0.1565176427496656518180806234654239164560069706202...

%t First[RealDigits[1/(Exp[2] - 1), 10, 100]]

%o (PARI) 1/(exp(2) - 1) \\ _Amiram Eldar_, Nov 08 2024

%Y Cf. A000796, A001113, A072334.

%Y Cf. A027641, A027642, A073747.

%K nonn,cons,easy

%O 0,2

%A _Paolo Xausa_, Nov 05 2024