A253191 Decimal expansion of log(2)^2.
4, 8, 0, 4, 5, 3, 0, 1, 3, 9, 1, 8, 2, 0, 1, 4, 2, 4, 6, 6, 7, 1, 0, 2, 5, 2, 6, 3, 2, 6, 6, 6, 4, 9, 7, 1, 7, 3, 0, 5, 5, 2, 9, 5, 1, 5, 9, 4, 5, 4, 5, 5, 8, 6, 8, 6, 6, 8, 6, 4, 1, 3, 3, 6, 2, 3, 6, 6, 5, 3, 8, 2, 2, 5, 9, 8, 3, 4, 4, 7, 2, 1, 9, 9, 9, 4, 8, 2, 6, 3, 4, 4, 3, 9, 2, 6, 9, 9, 0, 9, 3, 2, 7
Offset: 0
Examples
0.480453013918201424667102526326664971730552951594545586866864...
Links
- David H. Bailey, Jonathan M. Borwein and Richard E. Crandall, On the Khintchine Constant, Mathematics of Computation, Vol. 66, No. 217 (1997), pp. 417-431, see p. 419; alternative link, p. 4.
- Index entries for transcendental numbers
Programs
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Mathematica
RealDigits[Log[2]^2, 10, 103] // First
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PARI
log(2)^2 \\ Charles R Greathouse IV, Apr 20 2016
Formula
Integral_{0..1} log(1-x^2)/(x*(1+x)) dx = -log(2)^2.
Integral_{0..1} log(log(1/x))/(x+sqrt(x)) dx = log(2)^2.
Equals Sum_{k>=1} H(k)/(2^k * (k+1)) = 2 * Sum_{k>=1} (-1)^(k+1) * H(k)/(k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 05 2020
Equals Sum_{n >= 0} (-1)^n/(2^(n+1)*(n+1)^2*binomial(2*n+1,n)). See my entry in A002544 dated Apr 18 2017. Cf. A091476. - Peter Bala, Jan 30 2023
Equals 2*Integral_{x=-1..1} (abs(x)*log(x^2 + 1))/(x^2 + 1) dx. - Kritsada Moomuang, May 27 2025