I have the recursive sequence:
$$F_{n+1} = a _{\dot{}} 2^{-(F_n)^2}, F_0 = 1$$
and I want to redefine it as a function the same way the Fibonacci sequence can be rewritten using Binet's formula, but I quickly realized I'm in over my head. I attempted this using the proof of Binet's formula as a guide, but it isn't applicable to a sequence like this.
I decided to graph the sequence in Desmos to understand it better. I found the even terms of the sequence form one curve and the odd terms form a separate curve. If $a>\sqrt2$, then the two curves are mirrored logistic growths and converge to two separate points, but if $a<\sqrt2$, the two curves are exponential decays and seem to converge to a single point.
For the case of $a>\sqrt2$, is there any way to find a curve that models each of the two curves generated by the sequence? If not, is there a way to find the limit of the series as n approaches infinity for the odd and even terms?