Let the origin for x be at the position for spring A. Then the "stretch" of spring A is x and the "stretch" of spring B is 2L_0 - x. I am calling down the slope positive.
Set up your Newton's second to find a. Remember we have a weight term to consider here depending on the angle, but the spring forces don't. So we get
(Double check each of these to make sure I'm not making any mistakes. As I said, I'm running through this quickly.)
The right hand side is just a constant, so the particular solution will be in the form of x = C.
To get the period etc. solve the homogeneous equation
This will be a sine function of the form which you can get your period and amplitude information from.