A377813 Decimal expansion of arctanh(phi-1).

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

Offset

0,1

Comments

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

arctanh(phi-1) is the solution for real valued x in cosh(x) * sinh(x) = 1. - Colin Linzer, Nov 22 2024

Links

Table of n, a(n) for n=0..104.

Robert Frontczak, Problem H-884, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 59, No. 4 (2021), p. 373; Solution to Problem H-884, by Michel Bataille, ibid., Vol. 61, No. 2 (2023), pp. 187-188.

Index entries for transcendental numbers.

Formula

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

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

Equals arccosh(sqrt(phi)).

Equals arcsinh(sqrt(phi-1)).

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).

Equals 3*A202541. - Hugo Pfoertner, Nov 12 2024

Equals arcsinh(2)/2. - Colin Linzer, Nov 22 2024

Equals Sum_{k>=1} arccoth(phi^(2*k) - phi^(-2*k)), where phi is the golden ratio (A001622) (Frontczak, 2021). - Amiram Eldar, Dec 28 2025

Example

0.721817737589405171246638370136552634702...

Mathematica

RealDigits[ArcTanh[GoldenRatio - 1], 10, 120][[1]] (* Amiram Eldar, Nov 12 2024 *)

Prog

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

Crossrefs

Cf. A001622, A202541.

Sequence in context: A154020 A060991 A120455 * A268895 A394003 A394702

Adjacent sequences: A377810 A377811 A377812 * A377814 A377815 A377816

Keyword

nonn,cons

Author

Colin Linzer, Nov 08 2024

Extensions

More terms from Amiram Eldar, Dec 28 2025

Status

approved