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

%I #20 Jun 17 2023 03:17:49

%S 1,6,3,7,1,6,0,2,6,7,1,1,7,9,6,7,2,4,6,4,3,7,2,6,4,4,9,5,8,0,0,5,7,3,

%T 5,7,3,9,3,7,2,6,3,3,6,8,4,5,0,6,9,4,6,1,9,2,6,2,2,8,2,4,1,9,9,2,6,6,

%U 6,9,0,2,9,7,0,6,0,9,1,1,2,5,8,2,4,3,3,0

%N Decimal expansion of Sum_{i>=0} (1/(2*i^2+1)).

%H Ivan Panchenko, <a href="/A232883/b232883.txt">Table of n, a(n) for n = 1..1000</a>

%F Equals 1/2 + ( Pi/sqrt(2) ) /tanh( Pi/sqrt(2) ) /2.

%F Equals 1 + Sum_{k>=1} (-1)^(k+1)*zeta(2*k)/2^k. - _Amiram Eldar_, Jun 17 2023

%e 1.6371602671179672464372644958005735739372633684506946192622824199...

%t RealDigits[1/2 + Pi/Sqrt[2]/Tanh[Pi/Sqrt[2]]/2, 10, 90][[1]]

%o (PARI) sumnumrat(1/(2*x^2+1), 0) \\ _Charles R Greathouse IV_, Jan 20 2022

%o (PARI) 1/2 + Pi/sqrt(2)/tanh(Pi/sqrt(2))/2 \\ _Charles R Greathouse IV_, Jan 20 2022

%Y Cf. A072691 (Sum_{i>=1} (1/(2*i^2))), A113319 (Sum_{i>=0} (1/(i^2+1))).

%K nonn,cons

%O 1,2

%A _Bruno Berselli_, Dec 02 2013