Bariz - Problem

I suspect that the problem missed condition $a_i$ are positive integers, for then it is an classic problem. The idea is that the sequence is uniquely determined by its difference $a_i-a_{i-1} \in \{-1,0,1 \} , 1 \le i \le 99$ and minimum term $\min a_i$. Condition (i) says $a_i \in \{1,2,3 \}$, so there are $3 \cdot 3^{99}=3^{100}$ sequences with minimum term at most $3$. Among them, there are $2^{100}$ sequences with only $1$ and $2$, which should be discarded. So the desired number is $3^{100}-2^{100}$.