New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)

In math.CA/0211148 we observed that $\ln (4\mathrm{/}\pi )$ is an “alternating” analog of Euler’s constant $\gamma$. Here we use the binary expansion of an integer to give a rational series for $\ln (4\mathrm{/}\pi )$ 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.