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%I #35 Jan 04 2022 21:50:10
%S 6,6,1,5,7,0,1,9,2,0,7,3,5,8,5,1,1,2,0,4,4,5,7,3,8,9,2,8,4,6,0,7,9,3,
%T 9,5,2,1,7,6,4,2,4,6,6,5,8,9,5,5,6,9,7,9,8,6,9,1,9,8,4,8,5,4,5,0,1,8,
%U 9,5,0,9,7,9,4,2,6,0,1,7,2,0,7,5,9,5,8,8,8,7,7,9,1,1,8,6,9,3,7,2,4,4,9,2,7,9,4,8
%N Alternating sum of inverse of increasing integers with a difference of 0, 1, 2, 3, ...: 1 - 1/2 + 1/4 - 1/7 + 1/11 - 1/16 + 1/22 - 1/29 + 1/37 + ... i.e., alternating series based on A226985.
%F See Mathematica program. - _Joerg Arndt_, Sep 09 2013
%e 0.66157019207358511204457389...
%p c:= Sum( (-1)^k/(1+binomial(k+1, 2)), k=0..infinity):
%p Re(evalf(c, 120)); # _Alois P. Heinz_, Sep 09 2013
%t N[((-2 I) (LerchPhi[-1, 1, 1/2 - (I/2) Sqrt[7]] - LerchPhi[-1, 1, 1/2 + (I/2) Sqrt[7]]))/Sqrt[7], 99] (* _Joerg Arndt_, Sep 09 2013 *)
%t -(2*Im[PolyGamma[(1-I*Sqrt[7])/4] - PolyGamma[(3-I*Sqrt[7])/4]])/Sqrt[7] // RealDigits[#, 10, 100]& // First (* _Jean-François Alcover_, Sep 10 2013 *)
%o (PARI) default(realprecision,133); sumalt(k=1, 1/(1+k*(k-1)/2)*(-1)^(k+1))
%o (PARI) -(2*imag(psi((1-I*sqrt(7))/4)-psi((3-I*sqrt(7))/4)))/sqrt(7) \\ sumalt is faster; _Charles R Greathouse IV_, Sep 10 2013
%Y Cf. A226985, A000124.
%K nonn,cons
%O 0,1
%A _Didier Guillet_, Sep 08 2013
%E More terms from _Joerg Arndt_, Sep 09 2013