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

Abstract

We introduce the $2$-regular integer sequence A383066 $= (s(n))_{n \geq 1}$, which begins $0, 1, 1, 2, 3, 3, 2, \ldots$. We prove that the number of occurrences of an integer $m \geq 0$ 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.