Not an answer, but following the idea of Bullet and using the odd integers for $A$ instead gives the state diagram below for $k=2$ which can be used to find $\Delta^{(k)}_{i}$ (the directions at each level are given by $S_{n}$, the $n$-th digit from the right in the quaternary expansion of $i$, when the digits $S_n$ does not appear one terminates at the present node):
