A $2$-Regular Sequence That Counts The Divisors of $n^2+1$

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