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Let $P(n)$ be the number of polyominoes of $n$ cells and $\lambda$ beKlarner's constant, that is, $\lambda=\lim_{n\to\infty} \sqrt[n]{P(n)}$. Weshow 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 wellknown conjecture that there exist positive $C,\theta$ so that $P(n)\simCn^{-\theta}\lambda^n$ for every $n$. In fact, if we assume another popularconjecture 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 alsoconjecture that the ratio of the number of some class of polyominoes, namelyinconstructible polyominoes, over $P(n)$ is decreasing, by observing thisbehavior for the available values. The conjecture opens a nice approach tobounding $\lambda$ 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 likelyproperties of the function $P(n)$, instead of giving new insights of thestructure of polyominoes. The techniques can be applied to other latticeanimals and self-avoiding polygons of a given area with almost no change.

An asymptotic lower bound on the number of polyominoes