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A261622 Decimal expansion of the Dirichlet beta function at 1/3.

%I A261622 #18 Feb 16 2025 08:33:26
%S A261622 6,1,7,8,5,5,0,8,8,8,4,8,8,5,2,0,6,6,0,7,2,5,3,8,9,9,4,7,2,7,9,9,3,1,
%T A261622 6,5,7,1,0,6,2,3,5,4,7,8,9,9,3,8,6,5,0,0,2,2,5,5,1,5,2,8,2,2,9,5,6,0,
%U A261622 7,7,8,0,5,2,7,2,5,0,4,4,6,5,4,1,0,1,3,9,3,4,6,1,5,5,3,9,9,5,7,0,3,7,5,6,1
%N A261622 Decimal expansion of the Dirichlet beta function at 1/3.
%H A261622 G. C. Greubel, <a href="/A261622/b261622.txt">Table of n, a(n) for n = 0..10000</a>
%H A261622 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>
%H A261622 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_beta_function">Dirichlet beta function</a>
%F A261622 beta(1/3) = (zeta(1/3, 1/4) - zeta(1/3, 3/4))/2^(2/3).
%e A261622 0.6178550888488520660725389947279931657106235478993865002255152822956...
%p A261622 evalf(Sum((-1)^n/(2*n+1)^(1/3), n=0..infinity), 120); # _Vaclav Kotesovec_, Aug 27 2015
%t A261622 RealDigits[DirichletBeta[1/3],10,105]//First
%o A261622 (PARI) beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x
%o A261622 beta(1/3) \\ _Charles R Greathouse IV_, Oct 18 2024
%Y A261622 Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A261623 (beta(1/4)), A261624 (beta(1/5)).
%K A261622 cons,easy,nonn
%O A261622 0,1
%A A261622 _Jean-François Alcover_, Aug 27 2015