Question 184
Consider a sequence $(a_n)$: $a_1 = a, a_2 = b$, where $a$ and $b$ are positive integers, and for all $n \ge 2$ the term $a_{n+1}$ equals the number of indices $i$, $1 \le i \le n$, such that $a_i = a_n$.
Determine all pairs $(a, b)$ for which the following condition holds: there exists an index $n_0$ such that the sequence $(a_n + a_{n+1}), n \ge n_0$ is non-decreasing.
Determine all pairs $(a, b)$ for which the following condition holds: there exists an index $n_0$ such that the sequence $(a_n + a_{n+1}), n \ge n_0$ is non-decreasing.