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

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

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..20000

Travis Sherman, Summation of Glaisher- and Apéry-like Series, University of Arizona, 2000. (Scanned local copy without blank on p.12, with permission from the author.)

Index entries for transcendental numbers

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).

Example

0.2315648205771943924969290712315327600164063500492988708153012286...

Mathematica

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

Prog

(PARI) asinh(1/2)^2 \\ Michel Marcus, Dec 06 2020
(Magma)
m:= 250; SetDefaultRealField(RealField(m));
A339470:= Argsinh(1/2)^2;
Prune(Reverse(IntegerToSequence(Floor(( A339470 )*10^(Floor(m/2)) )))); // G. C. Greubel, Nov 29 2025
(SageMath)
A339470=numerical_approx(asinh(1/2)^2, digits= 250)
print([ZZ(i) for i in A339470.str()[1:-5] if i.isdigit()]) # G. C. Greubel, Nov 29 2025

Crossrefs

Cf. A001622, A002390, A002736, A086467.

Sequence in context: A195508 A389461 A049274 * A130508 A341635 A182938

Adjacent sequences: A339467 A339468 A339469 * A339471 A339472 A339473

Keyword

nonn,cons

Author

Robert Bilinski, Dec 06 2020

Status

approved