  We examine some patterns of Majorana neutrino mass matrix which is compatible
with the phenomenological lepton flavor mixing matrix and non-observation of
neutrinoless double beta decay. Imposing $(M_{\nu})_{ee}=0$ for the Majorana
neutrino mass matrix in the leading order, we obtain a relationship between the
solar mixing angle and the neutrino masses $m_1$ and $m_2$. Additional possible
texture zeros are assigned to the mass matrix so as for the nonvanishing
$\theta_{13}$ to be predictable in terms of neutrino masses. We also show how
three neutrino masses can be predicted from the solar mixing angle and the
experimental results of $\Delta m^2_{sol}$ and $\Delta m^2_{atm}$ in this
framework, and we discuss naturality of the forms of the mass matrix found in
our work.
