A307178 - OEIS

%I #38 Jan 26 2024 13:54:06

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

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

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

%N Decimal expansion of coth(1/2).

%C By the Lindemann-Weierstrass theorem, this constant is transcendental. - _Charles R Greathouse IV_, May 14 2019

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

%F Equals (exp(1)+1)/(exp(1)-1).

%F Equals (BesselI(3/2,1/2)/BesselI(1/2,1/2))+2.

%F Equals BesselI(-1/2,1/2)/BesselI(1/2,1/2).

%F Equals 2 * Sum_{k>=0} B(2*k)/(2*k)!, where B(2*k) = A000367(k)/A002445(k) are the Bernoulli numbers. - _Amiram Eldar_, Nov 25 2020

%F Equals 2 * A073333 + 1. - _Antonio Graciá Llorente_, Jan 21 2024

%e 2.163953413738... = 2 + 1/(6 + 1/(10 + 1/(14 + 1/(18 + ...)))).

%t RealDigits[Coth[1/2], 10, 120][[1]] (* or *) BesselI[-1/2, 1/2]/BesselI[1/2, 1/2]

%o (PARI) cotanh(1/2) \\ _Michel Marcus_, Mar 28 2019

%Y Cf. A016825 (continued fraction), A073333, A073747 (coth(1)).

%Y Cf. A000367, A002445.

%K cons,nonn

%O 1,1

%A _Terry D. Grant_, Mar 27 2019