Well, it turns out that you don't really need any high-level mathematics for this question, and I'm just wondering how come I didn't figure it out at that time, haha. So here's the whole set up again:
It'd be useful to see that the equilibrium position should have zero gravitational potential energy:
The trick to solving this question is to lift each pendulum up one after the other; so let's give the first pendulum a push and see what happens to the potential energy:
Notice that both pendulum bobs are raised by the same length, which explains the coefficient of '2'. Now let's give the lower pendulum another push and let's see what happens:
Notice that we've just added another term to the potential energy, and this time the coefficient is '1' because only one bob is raised.
Quite easy right? Haha.
Well, just click on it to enlarge it!