The only time I believe this should be done is if it is the first time you mow their lawn. What I mean is; if they flag you down while you are heading to one of your current customers and tell you they haven't been able to mow their lawn in a month.--then you double charge. Or if you get a call for a full maintenance account starting late in the season. When you arrive and their lawn is extremly high--you double charge the first visit. I don't think you will have a hard time explaining this to customers. When I take on customers that only want service on a bi-weekly plan, I factor that in and give them a price at least $5 higher than I normally would.
You should also keep in mind that mowing everyother week may hurt you. What if a neighbor is needing maintenance, and they see how out of control one of your bi-weekly lawns has gotten, you will have a hard time convencing this potential customer that you do solid work.