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

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, 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, 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

Offset: 0

Author

Jean-François Alcover, Aug 27 2015

Keywords

• cons
• easy
• nonn

Examples

			0.6178550888488520660725389947279931657106235478993865002255152822956...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's MathWorld, Dirichlet Beta Function
• Wikipedia, Dirichlet beta function

Crossrefs

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)).

Programs

Maple

evalf(Sum((-1)^n/(2*n+1)^(1/3), n=0..infinity), 120); # Vaclav Kotesovec, Aug 27 2015

Mathematica

RealDigits[DirichletBeta[1/3],10,105]//First

PARI

beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x
beta(1/3) \\ Charles R Greathouse IV, Oct 18 2024

Formula

beta(1/3) = (zeta(1/3, 1/4) - zeta(1/3, 3/4))/2^(2/3).