A377813 - OEIS

%I #21 Dec 01 2024 11:37:51

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

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

%U 5,8,0,1,4,1,2,3,3

%N Decimal expansion of arctanh(phi-1).

%C arctanh(phi-1) is the solution for real valued x in tanh(x) = d/dx tanh(x).

%C arctanh(phi-1) is the solution for real valued x in cosh(x) * sinh(x) = 1. - _Colin Linzer_, Nov 22 2024

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

%F Equals (1/2)*log(2+sqrt(5)).

%F Equals (3/2)*log(phi).

%F Equals arccosh(sqrt(phi)).

%F Equals arcsinh(sqrt(phi-1)).

%F Equals f(phi-1) with f(x) = (1/2)*log((2-x+2*sqrt(1-x))/x), a branch of the converse function of the derivative of tanh(x).

%F Equals 3*A202541. - _Hugo Pfoertner_, Nov 12 2024

%F Equals arcsinh(2)/2. - _Colin Linzer_, Nov 22 2024

%e 0.721817737589405171246638370136552634702...

%t RealDigits[ArcTanh[GoldenRatio - 1], 10, 120][[1]] (* _Amiram Eldar_, Nov 12 2024 *)

%o (PARI) atanh((1+sqrt(5))/2-1)

%Y Cf. A001622, A202541.

%K nonn,cons

%O 0,1

%A _Colin Linzer_, Nov 08 2024