Counting terms $U_n$ of third order linear recurrences with $U_n=u^2+nv^2$

TL;DR

This paper investigates the distribution of indices n for which a third order linear recurrence term U_n can be expressed as u^2 + n v^2, providing upper bounds for their count using advanced number theory techniques.

Contribution

It extends previous results on Fibonacci numbers to a broad class of third order linear recurrences, establishing new upper bounds for the count of such terms.

Findings

For large classes of ternary sequences, the number of such n is bounded by x/(log x)^{0.05}.

The method adapts techniques from Fibonacci sequence analysis to more general recurrences.

Provides a framework for counting special forms of recurrence sequence terms.

Abstract

Given a recurrent sequence ${\bf U}:=\{U_n\}_{n\ge 0}$ we consider the problem of counting ${\mathcal M}_U(x)$, the number of integers $n\le x$ such that $U_n=u^2+nv^2$ for some integers $u,v$. We will show that ${\mathcal M}_U(x)\ll x(\log x)^{-0.05}$ for a large class of ternary sequences. Our method uses many ingredients from the proof of Alba Gonz\'alez and the second author that ${\mathcal M}_F(x)\ll x(\log x)^{-0.06}$, with $\bf F$ the Fibonacci sequence.