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

%I #25 Aug 26 2018 12:41:34

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

%C 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 G. C. Greubel, <a href="/A262178/b262178.txt">Table of n, a(n) for n = 0..10000</a>

%F Equals (zeta(2, 1/6) - zeta(2, 2/3))/36, where zeta(s,a) is the Hurwitz zeta function.

%e 1 - 1/16 + 1/49 - 1/100 + 1/169 - 1/256 + 1/361 - 1/484 + ...

%e 0.9515177134164150418664828314727415315447285082326970513300324315296113...

%t RealDigits[(Zeta[2, 1/6] - Zeta[2, 2/3])/36, 10, 100][[1]]

%o (PARI) sumalt(k=0, (-1)^k/(3*k+1)^2) \\ _Michel Marcus_, Sep 14 2015

%o (PARI) zetahurwitz(2,1/6)/36 - zetahurwitz(2,2/3)/36 \\ _Charles R Greathouse IV_, Jan 31 2018

%Y Cf. A006752.

%Y Cf. A113476: Sum_{k>=0} (-1)^k/(3*k+1).

%Y Cf. A226735: Sum_{k>=0} (-1)^k/(3*k+1)^3.

%K nonn,cons

%O 0,1

%A _Bruno Berselli_, Sep 14 2015