New Vacca-Type Rational Series for Euler's Constant and Its   "Alternating" Analog ln(4/Pi)
  In math.CA/0211148 we observed that $\ln(4/ \pi)$ is an "alternating" analog of Euler's constant $\gamma$. Here we use the binary expansion of an integer to give a rational series for $\ln(4/ \pi)$ analogous to Vacca's series for $\gamma$. Also, using a known generalization of Vacca's series to other bases, we accelerate Addison's series for $\gamma$. Along the way, we give new proofs of Vacca's and Addison's formulas. 
Jonathan Sondow