2023 IOQM/Problem 10

The sequence $\{a_n\}_{n\geq0}$ is defined by $a_0 = 1$ , $a_1 = -4$ , and $a_{n+2} = -4a_{n+1} - 7a_n, \text{for } n \geq 0.$ Find the number of positive integer divisors of $a_{50}^2 - a_{49}a_{51}$ .