  Let $P(n)$ be the number of polyominoes of $n$ cells and $\lambda$ be
Klarner's constant, that is, $\lambda=\lim_{n\to\infty} \sqrt[n]{P(n)}$. We
show that there exist some positive numbers $A,T$, so that for every $n$ \[
P(n) \ge An^{-T\log n} \lambda^n. \] This is somewhat a step toward the well
known conjecture that there exist positive $C,\theta$ so that $P(n)\sim
Cn^{-\theta}\lambda^n$ for every $n$. In fact, if we assume another popular
conjecture that $P(n)/P(n-1)$ is increasing, we can get rid of $\log n$ to have
\[ P(n)\ge An^{-T}\lambda^n. \] Beside the above theoretical result, we also
conjecture that the ratio of the number of some class of polyominoes, namely
inconstructible polyominoes, over $P(n)$ is decreasing, by observing this
behavior for the available values. The conjecture opens a nice approach to
bounding $\lambda$ from above, since if it is the case, we can conclude that \[
\lambda < 4.1141, \] which is quite close to the current best lower bound
$\lambda > 4.0025$ and greatly improves the current best upper bound $\lambda <
4.5252$.
  The approach is merely analytically manipulating the known or likely
properties of the function $P(n)$, instead of giving new insights of the
structure of polyominoes. The techniques can be applied to other lattice
animals and self-avoiding polygons of a given area with almost no change.
