Consider the algorithm that starts with a

single square and on each of its n iterations adds new squares all around the

outside. How many one$-$by$-$one squares are there after n iterations? $[$Gar$99]$

$($In the parlance of cellular automata theory, the answer is the number of cells

in the von Neumann neighborhood of range n$.)$ The results for n $=0,1,$ and

$2$ are illustrated below