On the bottom page 13-13 of Volume III of "The Feynman Lectures on
Physics," Feynman states: "We leave it as a puzzle for you to find the
energy and amplitudes a[n] for this bound state." (For readers without
access to this book, there's too much background to state the full
question succinctly)

I'm trying to solve this puzzle, but I'm basically stuck. I'm not
necessarily looking for a full solution, but only some advice on how
to set up the problem.

He's asking for the energy of this bound state, but isn't the energy just
E = E0 + Sqrt(4A^2 + F^2)
as written on page 13-13?

And how do you setup the equations to discover the amplitudes?