  The number of independent sets is equivalent to the partition function of the
hard-core lattice gas model with nearest-neighbor exclusion and unit activity.
We study the number of independent sets $m_{d,b}(n)$ on the generalized
Sierpinski gasket $SG_{d,b}(n)$ at stage $n$ with dimension $d$ equal to two,
three and four for $b=2$, and layer $b$ equal to three for $d=2$. The upper and
lower bounds for the asymptotic growth constant, defined as
$z_{SG_{d,b}}=\lim_{v \to \infty} \ln m_{d,b}(n)/v$ where $v$ is the number of
vertices, on these Sierpinski gaskets are derived in terms of the results at a
certain stage. The numerical values of these $z_{SG_{d,b}}$ are evaluated with
more than a hundred significant figures accurate. We also conjecture the upper
and lower bounds for the asymptotic growth constant $z_{SG_{d,2}}$ with general
$d$.
