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%I A271563 #18 Apr 10 2016 19:21:13
%S A271563 3,4,4,5,4,3,6,3,6,7,9,2,3,7,0,6,4,0,3,3,2,0,5,3,3,8,7,9,0,0,2,0,4,3,
%T A271563 0,6,5,8,9,4,2,5,9,7,4,6,1,3,5,9,2,1,2,5,5,0,8,5,7,7,7,9,6,3,2,8,5,7,
%U A271563 8,3,4,4,2,8,6,2,5,2,7,6,2,1,0,4,5
%N A271563 Decimal expansion of Sum_{j>=0} Sum_{i>=0} (-1/4)^i*(-1)^j*binomial(2i,i)/((2j+1)(i+2j+2)).
%F A271563 Equals (Pi - 2*sqrt(1+sqrt(2)) * arctan(2*sqrt(2+10*sqrt(2))/7)) / sqrt(2). - _Vaclav Kotesovec_, Apr 10 2016
%e A271563 0.3445436367923706403320533879002043065894259746135921255085777...
%p A271563 evalf((Pi - 2*sqrt(1+sqrt(2)) * arctan(2*sqrt(2+10*sqrt(2))/7)) / sqrt(2), 120); # _Vaclav Kotesovec_, Apr 10 2016
%t A271563 RealDigits[(Pi - 2*Sqrt[1 + Sqrt[2]] * ArcTan[(2/7)*Sqrt[2 + 10*Sqrt[2]]])/Sqrt[2], 10, 120][[1]]
%t A271563 N[Sum[Sum[((-1)^(i + j) 4^-i Binomial[2 i, i])/((1 + 2 j) (2 + i + 2 j)), {i, 0, Infinity}], {j, 0, Infinity}]]
%o A271563 (PARI) (Pi - 2*sqrt(1+sqrt(2)) * atan(2*sqrt(2+10*sqrt(2))/7)) / sqrt(2)
%Y A271563 Decimal expansions of hypergeometric series: A244844, A263353, A263354, A263490, A263491, A263492, A263493, A263494, A263495, A263496, A263497, A263498, A229837, A268813, A086731, A002117, A076788.
%K A271563 cons,nonn
%O A271563 0,1
%A A271563 _John M. Campbell_, Apr 09 2016

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