A268086 Decimal expansion of Sum_{k>0} 1/(k*((k+1)^2+1)).

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, 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, 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

Offset: 0

Author

Bruno Berselli, Jan 26 2016

Keywords

• nonn
• cons

Comments

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.

Examples

			.297595969027714331872169889027156331536383020649824278231847237306809...
		

Crossrefs

Cf. A062158: numbers of the form k*((k+1)^2+1), with k>-2.

Cf. A268046: (1+i)*(H(1-i)-i*H(1+i))/4.

Programs

Maple

((1-I)*(harmonic(1-I) + I*harmonic(1+I)))/4:
Re(evalf(%, 106)); # Peter Luschny, Jan 27 2016

Mathematica

(1 - I)*(HarmonicNumber[1 - I] + I*HarmonicNumber[1 + I])/4 // Re // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Jan 26 2016 *)

Sage

# Warning: Floating point calculation. Adjust precision as needed
# and use some guard digits!
from mpmath import mp, chop, psi, coth, pi
mp.dps = 108; mp.pretty = True
chop((psi(0,I-1)-psi(0,1)-I+1)/2-pi*(I+1)*coth(pi)/4) # Peter Luschny, Jan 27 2016

Formula

Equals (1 - i)*(H(1-i) + i*H(1+i))/4, where H(z) is a harmonic number with complex argument.

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