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

TL;DR

This paper introduces a 2-regular sequence that counts the divisors of quadratic expressions, providing new insights into its properties and connections to Fibonacci numbers, along with a generating function for divisor counts.

Contribution

The paper defines a novel 2-regular sequence linked to divisor counts of quadratic forms and explores its properties and relationships to Fibonacci numbers.

Findings

Sequence counts divisors of m^2+1

Provides generating function for divisor counts

Establishes connection to Fibonacci sequence

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.