  In this work, we investigate Anderson localization of a one-dimensional
lattice with a mosaic off-diagonal quasi-periodic hopping. It is found that the
localization properties of zero-energy states depend sensitively on the parity
of mosaic modulation period $\kappa$. If the mosaic period is an odd integer,
there is no Anderson localization transition for an arbitrarily large
quasiperiodic hopping. The zero-energy state is a critical state. While for an
even number period and a generic quasiperiodic hopping, the zero-energy state
would be a localized edge state at left (or right)-hand end of system. The
above even-odd effects of mosaic period have been also transmitted into the
other eigenstates near zero-energy. More specifically, when mosaic period is
odd, there always exists an energy window in which the eigenstates are critical
for arbitrarily strong quasiperiodic hopping. While for even period, there
exists Anderson localization transition as the hopping strength increases. In
addition, the Lyapunov exponent $\gamma(E)$ and the mobility edges $E_c$ are
exactly obtained. With the Lyapunov exponent, we further find that there exist
critical regions in the parameter $\tau-E$ (for $\kappa=1$) and $\lambda-E$
(for $\kappa>1$) plane. Furthermore, when $E$ approaches mobility edges $E_c$,
it is found that the critical index of localization length is $\nu=1$. At the
end, we show that the systems with different $E$ can be characterized by
Lyapunov exponent $\gamma(E)$ and Avila acceleration $\omega(E)$.
