Numeric Springing (Game math-related question)

This blogpost describes how to use numeric springing. I was able to successfully implement this in my hobby project, however my limited math knowledge is not enough to dig dipper and derive the equation for finding the total time. To be more specific, given all the known variables (spring frequency, damping, time step, current and target positions etc) how do I find the total time that will take for a spring to come to rest. Unfortunately the author of the blogpost is unreachable, but since my question has to do with physics and game-math in general, I thought this is a great place to ask such a question.