For all positive integers, we can represent them uniquely in the system $\langle -1+i, Z_2 \rangle$. For example, $2$ can be represented as $(1100)_{-1+i}$. Given a number $k$, I'd like to ask how to determine the smallest positive integer $f(k)$ whose representation contains exactly $k$ times of $1$.
Here are some $f(k)$ for small $k$: $f(x)=x \ (1 \leq x \leq 7),\ f(8)=23,\ f(9)=39,\ f(10)=55,\ f(11)=71,\ f(12)=87$. But I haven't found out some general pattern of $f$.
Added: the generating function of $f$ seems to be $\frac{-15x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x}{16x^7 - 16x^6 - x + 1}$. I'm not sure whether it holds for all $k$.