Sum of Modes

For a **binary** string $T$, let $f(T) = $ the number of modes$^\dagger$ of string $T$. For example, $f(101) = 1$, as it has a unique mode $1$ while $f(01) = 2$, as it has $2$ modes, both $0$ and $1$. Given a **binary** string $S$ of length $N$, compute the value of $\displaystyle{\sum_{L = 1}^{N} \sum_{R = L}^{N} f(S[L,R])}$, where $S[L,R]$ denotes the substring $S_LS_{L+1}\ldots S_R$. $^\dag