We call a map $f:{\mathbb Z}\times {\mathbb Z} \to {\mathbb Z}$ a *Fibonacci grid* if for all $x,y \in {\mathbb Z}$ we have that $f(x,y)$ is the sum of the neighboring values, that is, $$f(x,y) = f(x-1,y)+f(x+1,y) + f(x,y-1) + f(x,y+1).$$
Are there uncountably many Fibonacci grids?



Fibonacci grids
Yes.


You may make two adjacent rows using any two strings of integers that you like:



> 
> ...$\ 3\ \ 1\ \ \ 4\ \ \ 1\ \ \ 5\ \ \ 9\ \ \ 2\ \ \ 6\ \ \ 5\ \ \ 3\ \ \ 5\ \ \ 8\ $...
> 
> 
> ...$\ 2\ \ 7\ \ \ 1\ \ \ 8\ \ \ 2\ \ \ 8\ \ \ 1\ \ \ 8\ \ \ 2\ \ \ 8\ \ \ 4\ \ \ 5\ $...
> 
> 
> 


Then fill in the rows above and below them one at a time; it's obvious that you can do this, and that you will fill it in with integers:



> 
> ...$ \ \ {}^-13\ \ 1 \ {}^-16 \ \ \ etc.$
> 
> 
> ...$\ 3\ \ 1\ \ \ 4\ \ \ 1\ \ \ 5\ \ \ 9\ \ \ 2\ \ \ 6\ \ \ 5\ \ \ 3\ \ \ 5\ \ \ 8\ $...
> 
> 
> ...$\ 2\ \ 7\ \ \ 1\ \ \ 8\ \ \ 2\ \ \ 8\ \ \ 1\ \ \ 8\ \ \ 2\ \ \ 8\ \ \ 4\ \ \ 5\ $...
> 
> 
> ...$\ \ \ \ \, 3\, {}^-18\ \ 4\ \ \ \ etc. $
> 
> 
> 


