SILK-ROAD 2015 Problem 3

Let $B_n$ be the set of all sequences of length $n$ , consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$\varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} (0 \leq i \leq n-1).$$ . Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)}$ . .