In this work we prove the lower bound for the number of T-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, T-periodic in time, with T-Maslov indices i_0, i_∞ at the origin and at infinity, has at least |i_∞ - i_0| periodic solutions, and an additional one if i_0 is even. Our argument combines the Poincaré-Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.
