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A113476 Decimal expansion of (log(2) + Pi/sqrt(3))/3.

%I A113476 #57 Mar 07 2025 06:26:04
%S A113476 8,3,5,6,4,8,8,4,8,2,6,4,7,2,1,0,5,3,3,3,7,1,0,3,4,5,9,7,0,0,1,1,0,7,
%T A113476 6,6,7,8,6,5,2,2,1,2,7,4,8,4,3,3,1,9,4,3,2,3,0,1,8,8,3,1,4,9,6,0,5,0,
%U A113476 5,6,0,1,0,3,2,0,1,6,1,9,9,7,6,3,3,2,9,4,3,8,4,0,2,8,2,6,2,8,5,4,6,6,0,7
%N A113476 Decimal expansion of (log(2) + Pi/sqrt(3))/3.
%C A113476 This number is transcendental - this follows from a result of Baker (1968) on linear forms of algebraic numbers.
%D A113476 Jolley, Summation of Series, Dover (1961), eq (79) page 16.
%D A113476 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, equation 21.16
%H A113476 Ivan Panchenko, <a href="/A113476/b113476.txt">Table of n, a(n) for n = 0..1000</a>
%H A113476 A. Baker, <a href="http://dx.doi.org/10.1112/S0025579300002588">Linear forms in the logarithms of algebraic numbers (IV)</a>. Mathematika, 15 (1968) pp. 204-216
%H A113476 L. Euler, <a href="http://eulerarchive.maa.org/">De fractionibus continuis observationes</a>, The Euler Archive, Index Number 123, Section 5
%H A113476 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/EulersSeriesTransformation.html">Euler's Series Transformation</a>.
%H A113476 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A113476 Equals Integral_{x = 0..1} dx/(1+x^3) = Sum_{k >= 0} (-1)^k/(3*k+1) = 1 - 1/4 + 1/7 - 1/10 + 1/13 - 1/16 + ... (see A016777). - _Benoit Cloitre_, _Alonso del Arte_, Jul 29 2011
%F A113476 Generalized continued fraction: 1/(1 + 1^2/(3 + 4^2/(3 + 7^2/(3 + 10^2/(3 + ... ))))) due to Euler. For a sketch proof see A024217. - _Peter Bala_, Feb 22 2015
%F A113476 Equals (1/2)*Sum_{n >= 0} n!*(3/2)^n/(Product_{k = 0..n} 3*k + 1) = (1/2)*Sum_{n >= 0} n!*(3/2)^n/A007559(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(3*k + 1)). - _Peter Bala_, Dec 01 2021
%F A113476 From _Peter Bala_, Mar 03 2024: (Start)
%F A113476 Equals hypergeom([1/3,  1], [4/3], -1).
%F A113476 Gauss's continued fraction: 1/(1 + 1/(4 + 3^2/(7 + 4^2/(10 + 6^2/(13 + 7^2/(16 + 9^2/(19 + 10^2/(22 + 12^2/(25 + 13^2/(28 + ... )))))))))). (End)
%F A113476 Equals (1/12) * Sum_{n >= 0} (-1/2)^n * (9*n + 7)/((3*n + 2)*(n + 1)*binomial(2*n+1/3, n+1)). - _Peter Bala_, Mar 05 2025
%e A113476 0.835648848264721053337... = A073010 + A193535.
%t A113476 RealDigits[(Log[2]+\[Pi]/Sqrt[3])/3,10,120][[1]]  (* _Harvey P. Dale_, Mar 26 2011 *)
%o A113476 (PARI) 1/3*(log(2)+Pi/sqrt(3))
%Y A113476 Cf. A007559, A016777, A024217, A073010, A193534, A193535.
%K A113476 cons,nonn
%O A113476 0,1
%A A113476 _Benoit Cloitre_, Jan 08 2006