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A193535 Decimal expansion of log(2)/3.

%I A193535 #34 Feb 28 2024 01:38:41
%S A193535 2,3,1,0,4,9,0,6,0,1,8,6,6,4,8,4,3,6,4,7,2,4,1,0,7,0,7,1,5,2,7,2,5,5,
%T A193535 2,2,6,9,1,8,3,3,3,7,8,1,2,0,0,8,5,0,8,4,7,0,6,8,9,3,3,3,6,4,9,7,7,9,
%U A193535 7,8,7,3,9,8,9,8,9,8,2,3,8,5,3,5,2,8,7,7,7,5,6,6,5,4,7,2,8
%N A193535 Decimal expansion of log(2)/3.
%C A193535 This number is involved as an addend or subtrahend in the closed forms of certain series of reciprocals of integers (see for example A113476).
%D A193535 L. B. W. Jolley, Summation of Series, Dover (1961).
%D A193535 Murray R. Spiegel, Seymour Lipschutz, John Liu. Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill (2009): p. 135, equations 21.16 and 21.18.
%H A193535 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A193535 Equals lim_{n->oo} [Sum_{i = 1..n} i^2/(n^3 + i^3)]. [Jolley eq 292, p.52]
%F A193535 Equals Sum_{n>=1} (-1)^(n-1)/(n*2^n*binomial(2*n, n)). - _Arkadiusz Wesolowski_, Jan 20 2013
%F A193535 From _Amiram Eldar_, Aug 05 2020: (Start)
%F A193535 Equals Integral_{x=1..oo} 1/(x^4 + x) dx.
%F A193535 Equals Integral_{x=0..oo} 1/(exp(2*x) + 3) dx. (End)
%F A193535 From _Peter Bala_, Feb 27 2024: (Start)
%F A193535 Equals (1/2)*Sum_{k >= 0} (-1)^k/((3*k + 1)*(3*k + 2)) = (1/2)*(1/(2 + (1*2)^2/(18 + (4*5)^2/(2*18 + (7*8)^2/(3*18 + (10*11)^2/(4*18 +  ... )))))) (continued fraction). See A052502.
%F A193535 Equals 7/32 + (3/2)*Sum_{k >= 0} (-1)^k/((3*k + 1)*(3*k + 2)*(3*k + 3)*(3*k + 4)*(3*k + 5)). (End)
%e A193535 0.231049060186648...
%t A193535 RealDigits[(Log[2]/3), 10, 100][[1]]
%o A193535 (PARI) log(2)/3 \\ _Charles R Greathouse IV_, Jul 29 2011
%Y A193535 Cf. A052502, A113476.
%K A193535 nonn,cons
%O A193535 0,1
%A A193535 _Alonso del Arte_, Jul 29 2011