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%I #46 Jan 15 2021 17:54:23
%S 1,2,7,9,7,2,8,7,4,2,2,8,1,8,9,6,8,3,3,6,4,7,2,7,5,7,0,1,5,0,7,6,3,0,
%T 6,7,2,2,6,2,6,0,3,6,7,5,0,7,5,7,8,2,6,1,9,3,0,6,8,3,0,5,8,8,1,6,9,3,
%U 0,6,6,0,7,2,2,1,3,6,4,9,0,6,6,2,1,1,5,3,2,9,9,0,5,3,5,3,2,2,7,3,7,1,9,7,1,3,2,9,2,3
%N Decimal expansion of K = Sum_{m>=0} 1/(1 + 2*m + 4*m^2).
%C This constant K and the constant J = A339135 allow the expression of the real and imaginary parts of:
%C Psi(1/4 + i*sqrt(3)/4) = - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*sqrt(3)*K;
%C Psi(-1/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3) - sqrt(3)*K + Pi*tanh(Pi*sqrt(3)/2));
%C Psi(3/4 + i*sqrt(3)/4)= - J - i*sqrt(3)*k - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/4) + i*Pi*tanh(Pi*sqrt(3)/2).
%C Psi(-3/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3)/3 + sqrt(3)*K).
%C where Psi is the digamma function and i=sqrt-1).
%F Equals -i/(2*sqrt(3)) * (Psi(1/4 + i*sqrt(3)/4) - Psi(1/4 - i*sqrt(3)/4)).
%F Equals Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/3 - Sum_{m>=0} 1/(3 + 6*m + 4*m^2).
%e 1.27972874228189683364727570150763067226260...
%p K:= Re(sum(1/(1+2*n+4*n^2), n=0..infinity)):
%p evalf(K, 120); # _Alois P. Heinz_, Dec 06 2020
%t RealDigits[N[Re[Sum[1/(1 + 2*n + 4*n^2), {n, 0, Infinity}]], 110]][[1]]
%o (PARI) sumpos(n=0, 1/(1+2*n+4*n^2)) \\ _Michel Marcus_, Nov 28 2020
%Y Cf. A054569 (terms), A339135.
%K nonn,cons
%O 1,2
%A _Artur Jasinski_, Nov 28 2020
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