  We have studied the low-temperature behavior of the magnetization and the
specific heat of the bilayered perovskite system $%
La_{2-2x}Sr_{1+2x}Mn_{2}O_{7}$, for $0.30\leq x\leq 0.50$. Our analysis reveals
that below 30 K the temperature dependence of the magnetization of the
ferromagnetic samples, $x=0.30$, 0.32, and 0.36, in a field of 1T can be
interpreted in terms of the thermal excitations of a two-dimensional gas of
ferromagnetic magnons. The specific heat in zero field for these samples as for
the $x=0.50$ antiferromagnetic one, is linear with temperature between the
range of 1.8K $\leq $ T $\leq $ 10K. This behavior can be also explained by the
magnon gas model. By comparing specific heat measurements in zero field with
those taken in a field of 9T we are able to extract the lattice and electronic
contributions and determine the in-plane exchange interactions. That are found
to be in reasonable agreement with the values inferred from the analysis of the
magnetization data and also with the values reported by inelastic neutron
scattering studies. In addition, we found that the electronic density of states
obtained for the $x=0.50$ sample is in agreement with previous band structure
calculations.
