Question 156
For a word $S$ written in letters $A$ and $B$, let $f(S)$ denote the maximum number of non-intersecting $ABA$ subwords in $S$. For example, $f(ABBABBA) = 0$, $f(ABABABBA) = 1$ and $f(ABABABA) = 2$.
For $n = 4k + 1$, find the sum $\sum f(S)$ where $S$ runs over all $n$ letter words written in letters $A$ and $B$.
(Nyamdavaa Amar)
For $n = 4k + 1$, find the sum $\sum f(S)$ where $S$ runs over all $n$ letter words written in letters $A$ and $B$.
(Nyamdavaa Amar)