We introduce the 2-regular integer sequence A383066 = (s(n))_{n ≥ 1}, which begins 0, 1, 1, 2, 3, 3, 2, ... . We prove that the number of occurrences of an integer m ≥ 0 in this sequence is equal to τ(m²+1), the number of divisors of m² + 1. Using this fact, we give a generating function for τ(m²+1). We also discuss other interesting properties of s(n), including its relationship to the Fibonacci sequence.