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%I A256576 #12 Jun 13 2024 13:34:20
%S A256576 1,6,2,2,7,1,9,3,9,4,7,1,4,8,3,3,9,0,7,1,5,3,5,9,5,5,1,8,0,8,0,7,1,2,
%T A256576 0,6,4,7,3,4,9,9,7,5,1,4,0,0,3,4,6,3,1,4,2,5,8,8,6,7,2,7,2,9,3,7,8,1,
%U A256576 1,7,2,1,2,1,0,5,0,3,9,7,1,4,2,5,2,4,0,5,3,8,0,7,9,6,7,4,9,8,9,2
%N A256576 Decimal expansion of Ramanujan's constant G(1) = Sum_{r>=1} 1/(2r^3)*(1 + 1/3 + ... + 1/(2r-1)).
%H A256576 Marc-Antoine Coppo and Bernard Candelpergher, <a href="http://dx.doi.org/10.1016/j.jnt.2014.11.007">Inverse binomial series and values of Arakawa-Kaneko zeta functions</a>, J. Number Theory 150 (2015) 98-119 eq. (25).
%H A256576 R. Sitaramachandrarao, <a href="http://dx.doi.org/10.1016/0022-314X(87)90012-6">A Formula of S. Ramanujan </a>, Journal of Number Theory 25, 1-19 (1987)
%F A256576 G(1) = Pi^4/480 - (1/4)*Sum_{r>=2} (-1)^r*H(r-1,2)/r^2, where H(n,m) is the n-th harmonic number of order m.
%F A256576 F(1) = 7*Zeta(3)/16 = Sum_{r>=1} (1+1/3+1/5+..+1/(2r-1))/r^2 = 0.525899895... [Coppo] - _R. J. Mathar_, Jun 13 2024
%e A256576 0.1622719394714833907153595518080712064734997514...
%t A256576 digits = 100; G1 = NSum[(1/(2*r)^3)*(1/2*(EulerGamma + Log[4]) + 1/2*PolyGamma[ 1/2+r]), {r, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> 40, Method -> {"NIntegrate", "MaxRecursion" -> 10}]; RealDigits[G1, 10, digits] // First
%K A256576 nonn,cons
%O A256576 0,2
%A A256576 _Jean-François Alcover_, Apr 02 2015

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