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%I #21 Mar 19 2020 16:40:26
%S 2,9,7,5,9,5,9,6,9,0,2,7,7,1,4,3,3,1,8,7,2,1,6,9,8,8,9,0,2,7,1,5,6,3,
%T 3,1,5,3,6,3,8,3,0,2,0,6,4,9,8,2,4,2,7,8,2,3,1,8,4,7,2,3,7,3,0,6,8,0,
%U 9,2,9,6,8,0,9,3,1,7,6,5,1,2,8,8,4,2,6,1,1,0,5,1,3,9,0,2,4,6,4,7
%N Decimal expansion of Sum_{k>0} 1/(k*((k+1)^2+1)).
%C Also, decimal expansion of Integral_{x=0..1} (2 - (1-i)*x^(1-i) - (1+i)*x^(1+i))/(4 - 4*x) dx, where i is the imaginary unit.
%F Equals (1 - i)*(H(1-i) + i*H(1+i))/4, where H(z) is a harmonic number with complex argument.
%F Equals (Psi(i-1)-Psi(1)-i+1)/2-Pi*(i+1)*coth(Pi)/4), where Psi(x) is the digamma function. - _Peter Luschny_, Jan 27 2016
%e .297595969027714331872169889027156331536383020649824278231847237306809...
%p ((1-I)*(harmonic(1-I) + I*harmonic(1+I)))/4:
%p Re(evalf(%, 106)); # _Peter Luschny_, Jan 27 2016
%t (1 - I)*(HarmonicNumber[1 - I] + I*HarmonicNumber[1 + I])/4 // Re // RealDigits[#, 10, 100]& // First (* _Jean-François Alcover_, Jan 26 2016 *)
%o (Sage)
%o # Warning: Floating point calculation. Adjust precision as needed
%o # and use some guard digits!
%o from mpmath import mp, chop, psi, coth, pi
%o mp.dps = 108; mp.pretty = True
%o chop((psi(0,I-1)-psi(0,1)-I+1)/2-pi*(I+1)*coth(pi)/4) # _Peter Luschny_, Jan 27 2016
%Y Cf. A062158: numbers of the form k*((k+1)^2+1), with k>-2.
%Y Cf. A268046: (1+i)*(H(1-i)-i*H(1+i))/4.
%K nonn,cons
%O 0,1
%A _Bruno Berselli_, Jan 26 2016
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