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set of all sequences of length n consisting of zeros and ones, sum related

Source: SRMC 2015

September 2, 2018
SequenceStringsSumcombinatorics

Problem Statement

Let BnB_n be the set of all sequences of length nn, consisting of zeros and ones. For every two sequences a,b∈Bna,b \in B_n (not necessarily different) we define strings ε0ε1ε2…εn\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n and δ0δ1δ2…δn\delta_0\delta_1\delta_2 \dots \delta_n such that ε0=δ0=0\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)=ε0+ε1+ε2+⋯+εnw(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n . Find f(n)=∑a,b∈Bnw(a,b)f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} . .
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