Question 14
Given a positive integer $n$, every nonzero complex number of the form $r+si$, where $r$ and $s$ are integers, has a unique expansion in base $-n+i$ using the integers $0,1,\ldots,n^2$ as digits. That is,
$$\begin{aligned}r+si={}&a_m(-n+i)^m\\&+a_{m-1}(-n+i)^{m-1}\\&+\cdots+a_1(-n+i)+a_0\end{aligned}$$
for a unique nonnegative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from $\{0,1,2,\ldots,n^2\}$, with $a_m\ne0$. The number zero is represented by the single digit $0$. We write
$$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$$
to denote this base expansion. There are only finitely many integers $k+0i$ that have four-digit expansions
$$k=(a_3a_2a_1a_0)_{-3+i},\qquad a_3\ne0.$$
Find the sum of all such integers $k$.
$$\begin{aligned}r+si={}&a_m(-n+i)^m\\&+a_{m-1}(-n+i)^{m-1}\\&+\cdots+a_1(-n+i)+a_0\end{aligned}$$
for a unique nonnegative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from $\{0,1,2,\ldots,n^2\}$, with $a_m\ne0$. The number zero is represented by the single digit $0$. We write
$$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$$
to denote this base expansion. There are only finitely many integers $k+0i$ that have four-digit expansions
$$k=(a_3a_2a_1a_0)_{-3+i},\qquad a_3\ne0.$$
Find the sum of all such integers $k$.