A339470 Decimal expansion of log(phi)^2, where phi is the golden ratio (A002390^2).

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

Offset: 0

Author

Robert Bilinski, Dec 06 2020

Keywords

• nonn
• cons

Examples

			0.2315648205771943924969290712315327600164063500492988708153012286...
		

Links

• Travis Sherman, Summation of Glaisher- and Apéry-like Series, University of Arizona, 2000.
• Index entries for transcendental numbers

Crossrefs

Cf. A002390, A001622, A086467.

Programs

Mathematica

RealDigits[Log[GoldenRatio]^2, 10, 100][[1]] (* Amiram Eldar, Dec 06 2020 *)

PARI

asinh(1/2)^2 \\ Michel Marcus, Dec 06 2020

Formula

Equals arcsinh(1/2)^2 = A002390^2.

Equals (1/2)*Sum_{k>=1} ((k!)^2*(-1)^(k+1))/((2*k)!*k^2) = A086467/2.

Equals (1/3)*(zeta(2) - Sum_{k>=1} ((k!)^2*(-1)^k)/((2*k)!*(2*k+1)^2)).

Equals (1/2)*Sum_{k>=1} (-1)^(k+1)/A002736(k).