We have the following: - $b(n)=1$ for $n$ between 1 and 3 . - $b(n)=3$ for $n$ between $4^{2}-3 \cdot 4=4$ and $3 \cdot 4^{2}+3=51$. (Since $a \cdot 4^{2}-b \cdot 4+c$ takes on $3 \cdot 4 \cdot 4$ distinct values over $1 \leq a \leq 3,0 \leq b \leq 3,0 \leq c \leq 3$, with minimum 4 and maximum 51.) - $b(n)=5$ for $n$ between $4^{4}-3 \cdot 4^{3}-3 \cdot 4=52$ and $3 \cdot 4^{4}+3 \cdot 4^{2}+3=819$. - $b(n)=7$ for $n$ between $4^{6}-3 \cdot 4^{5}-3 \cdot 4^{3}-3 \cdot 4^{1}=820$ and $3 \cdot 4^{6}+3 \cdot 4^{4}+3 \cdot 4^{2}+3>2013$. Thus $$\sum_{i=1}^{2013} b(i)=7(2013)-2(819+51+3)=14091-2(873)=14091-1746=12345$$