Maximum density of an induced 5-cycle is achieved by an iterated blow-up of a 5-cycle

Let $C(n)$ denote the maximum number of induced copies of 5-cycles in graphs on $n$ vertices. For $n$ large enough, we show that $C(n)=a\cdot b\cdot c \cdot d \cdot e + C(a)+C(b)+C(c)+C(d)+C(e)$, where $a+b+c+d+e = n$ and $a,b,c,d,e$ are as equal as possible. Moreover, if $n$ is a power of 5, we show that the unique graph on $n$ vertices maximizing the number of induced 5-cycles is an iterated blow-up of a 5-cycle.