Solve this functional equation:

$$\frac{F(x)}{F(1)} = 2F\left(\frac{(1+x)^2}{4a}\right)-F(x/a)$$

for $F(x)$ where $a > 0$ is a parameter. I know there is a trivial constant solution, $F(x) = 1$. Is there a non-constant solution?

Assume that $F(x)$ is continuous, and all the smoothness properties that you require to arrive to an answer. Note that I would accept the answer in the form of an infinite product or sum, or integral ....

I do not know if it has an analytical solution, and have no reason to expect that it does. It showed up in some calculations I was doing. But if there is a technique I can apply, it would be nice to know about it.