Partly inspired by this question: I'm interested in the number $A_{m, n}$ of $2$-colorings of an $m \times n$ grid, say red and blue, such that no two (vertically or horizontally) adjacent squares are both blue. I doubt this has a closed formula for general $m, n$, so I'm more interested in asymptotics.
We can define $a_m = \lim_{n \to \infty} (A_{m, n})^{1/n}$, so, for example, by standard methods for linear recurrences we have $a_1 = \phi = \frac{1+\sqrt{5}}{2}$ (since $A_{1, n}$ are the Fibonacci numbers), $a_2 = 1 + \sqrt{2}$, and $a_3 \approx 3.631$. In general, one can compute $a_m$ by taking the largest eigenvalue of the $A_{m, 1} \times A_{m, 1}$ "adjacency" matrix indicating whether two colorings of the $m \times 1$ grid can be placed next to each other.
One can also show that the limit $a = \lim_{m \to \infty} a_m^{1/m}$ exists, and satisfies $a \leq a_m^{1/m}$ for each $m$, so for example $a \leq a_3^{1/3} \approx 1.537$. In the other direction, we see $A_{m, n} \geq 2^{mn/2}$ (by taking all colorings in which only those squares in an alternating checkerboard pattern are allowed to be blue), so $a \geq \sqrt{2}$.
My question is: what is $a$? Does $a = \sqrt{2}$? Is there a closed expression for $a$, e.g. as an infinite series? Alternately, are there general asymptotics for $A_{m, n}$ rather than just for fixed $m$? I would particularly be interested in asymptotics for $A_{n, n}$.
$A_{m,n}$ is OEIS A089934. $A_{n,n}$ is OEIS A006506, which mentions "Limit n ->infty (a(n))^(1/n^2) =c1=1.50304... is the hard square entropy constant"