AoPS - 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

$3^{100}-2^{100}$ .