A156057 Decimal expansion of log(3)/2.

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

Offset

0,1

Comments

Culler & Shalen show a bound of log(3)/2 on maximal injectivity under certain circumstances, see links.

Equals arctanh(1/2), the rapidity of an object traveling at half the speed of light. - Sean Stroud, May 13 2019

Links

Table of n, a(n) for n=0..98.

Marc Culler and Peter B. Shalen, Betti numbers and injectivity radii, Proceedings of the American Mathematical Society, Vol. 137, No. 11 (2009), pp. 3919-3922; preprint, arXiv:0902.0014 [math.GT], 2009.

R. S. Melham and A. G. Shannon, Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.

Michael Penn, This gnarly integral is actually easy??, YouTube video, 2023.

Index entries for transcendental numbers

Formula

Equals arctanh(1/2) = arccoth(2) = Integral_{x>2} 1/(x^2-1) dx. - Jean-François Alcover, Jun 04 2013

From Amiram Eldar, Aug 05 2020: (Start)

Equals Sum_{k>=0} 1/((2*k+1) * 2^(2*k+1)).

Equals Integral_{x=0..oo} 1/(exp(x) + 2) dx. (End)

Equals Sum_{k>=1} arctanh(1/Fibonacci(2*k+2)) (Melham and Shannon, 1995). - Amiram Eldar, Jan 15 2022

log(3)/2 = Sum_{n >= 1} 1/(n*P(n, 2)*P(n-1, 2)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(3)/2 = 0.54930614433(10...), correct to 11 decimal places. - Peter Bala, Mar 16 2024

Example

0.54930614433405484569762261846...

Mathematica

RealDigits[Log[3]/2, 10, 120][[1]] (* Harvey P. Dale, Apr 13 2016 *)

Prog

(PARI) log(3)/2 \\ Charles R Greathouse IV, May 15 2019

Crossrefs

Cf. A000045, A002391 (decimal expansion of natural logarithm of 3).

Sequence in context: A097943 A241420 A077142 * A125057 A021186 A195705

Adjacent sequences: A156054 A156055 A156056 * A156058 A156059 A156060

Keyword

nonn,cons,easy

Author

Jonathan Vos Post, Feb 03 2009

Extensions

All digits were wrong. Corrected by N. J. A. Sloane, Feb 05 2009

Offset 0 from Michel Marcus, May 13 2019

Status

approved