On paper [A procedure for improving the upper bound
for the number of $n$-ominoes](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CA6D64DA91090FFAA49EBC83C098E3AF/S0008414X00050628a.pdf/a-procedure-for-improving-the-upper-bound-for-the-number-of-n-ominoes.pdf) by D. A. Klarner & R. L. Rivest, it is known that the number $t(n)$ of polyominoes with area $n$ satisfies $3.72^n<t(n)<6.75^n$.
What is the best current result?



Number of polyominoes with area $n$
Wikipedia suggests that no-one has improved on Klarner, D.A.; Rivest, R.L. (1973) [A procedure for improving the upper bound for the number of n-ominoes](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CA6D64DA91090FFAA49EBC83C098E3AF/S0008414X00050628a.pdf/a-procedure-for-improving-the-upper-bound-for-the-number-of-n-ominoes.pdf), Canadian J of Math. 25 (3): 585–602, which gives $$\lim\_{n \to \infty} (t(n))^{1/n} \le 4.649551$$


However, that is not conclusive because it appears to overlook Barequet, G., Rote, G., & Shalah, M. (2016) [λ> 4: An improved lower bound on the growth constant of polyominoes](https://geometry.stanford.edu/papers/brs-lg4iubgcp-16/brs-lg4iubgcp-16.pdf), CACM, 59(7), 88-95, which gives a lower bound of $$t(n) \ge 4.00253176^n$$


Two of the same authors (Gill Barequet, Mira Shalah) have a [preprint on arxiv](https://arxiv.org/abs/1906.11447) which purports to show that the upper bound is $$t(n) \le 4.5252^n$$


