In $d$-dimensional lattice, we define a set $S_0$ be the zero point.
At step $i\geq 1$:
For each point $p\in S_{i-1}$, we can choose a single point $q$ who is a neighbour of $p$, and add $q$ into $s_{i-1}$ to obtain $S_i$.
Can we have a closed form for
$a_i=\max |S_i|$
for each $i$?
How about $d=2$?
It is clear that $a_1=2, a_2=4,a_3=8,a_4=16$ for any $d\geq 2$.
