Question 96
Let $a_1, a_2, \dots$ and $b_1, b_2, \dots$ be two sequences consisting of positive integers such that, for any positive integer $n$,
$$
(a_{n+1}, b_{n+1}) = \left( \frac{a_n}{2}, b_n + \frac{a_n}{2} \right) \text{ or } (a_{n+1}, b_{n+1}) = \left( a_n + \frac{b_n}{2}, \frac{b_n}{2} \right)
$$
holds. How many initial pairs $(a_1, b_1)$ with $1 \le a_1, b_1 \le 40$ are possible?
$$
(a_{n+1}, b_{n+1}) = \left( \frac{a_n}{2}, b_n + \frac{a_n}{2} \right) \text{ or } (a_{n+1}, b_{n+1}) = \left( a_n + \frac{b_n}{2}, \frac{b_n}{2} \right)
$$
holds. How many initial pairs $(a_1, b_1)$ with $1 \le a_1, b_1 \le 40$ are possible?