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A262178 Decimal expansion of Sum_{k>=0} (-1)^k/(3*k+1)^2.

%I A262178 #25 Aug 26 2018 12:41:34
%S A262178 9,5,1,5,1,7,7,1,3,4,1,6,4,1,5,0,4,1,8,6,6,4,8,2,8,3,1,4,7,2,7,4,1,5,
%T A262178 3,1,5,4,4,7,2,8,5,0,8,2,3,2,6,9,7,0,5,1,3,3,0,0,3,2,4,3,1,5,2,9,6,1,
%U A262178 1,3,4,3,0,2,2,7,5,8,3,0,2,1,9,9,3,4,7,4,8,9,3,7
%N A262178 Decimal expansion of Sum_{k>=0} (-1)^k/(3*k+1)^2.
%C A262178 Also, decimal expansion of Sum_{h>=0} Sum_{j=0..h} (-1)^j*binomial(h, j)/(4*(1 + h)*(1 + 6*j)*(2 + 3*j)).
%H A262178 G. C. Greubel, <a href="/A262178/b262178.txt">Table of n, a(n) for n = 0..10000</a>
%F A262178 Equals (zeta(2, 1/6) - zeta(2, 2/3))/36, where zeta(s,a) is the Hurwitz zeta function.
%e A262178 1 - 1/16 + 1/49 - 1/100 + 1/169 - 1/256 + 1/361 - 1/484 + ...
%e A262178 0.9515177134164150418664828314727415315447285082326970513300324315296113...
%t A262178 RealDigits[(Zeta[2, 1/6] - Zeta[2, 2/3])/36, 10, 100][[1]]
%o A262178 (PARI) sumalt(k=0, (-1)^k/(3*k+1)^2) \\ _Michel Marcus_, Sep 14 2015
%o A262178 (PARI) zetahurwitz(2,1/6)/36 - zetahurwitz(2,2/3)/36 \\ _Charles R Greathouse IV_, Jan 31 2018
%Y A262178 Cf. A006752.
%Y A262178 Cf. A113476: Sum_{k>=0} (-1)^k/(3*k+1).
%Y A262178 Cf. A226735: Sum_{k>=0} (-1)^k/(3*k+1)^3.
%K A262178 nonn,cons
%O A262178 0,1
%A A262178 _Bruno Berselli_, Sep 14 2015