Counting terms $U_n$ of third order linear recurrences with $U_n=u^2+n{v}^2$

Abstract

Given a recurrent sequence $\mathbf{U}:=\{U_n{\}}_{n\ge 0}$ we consider the problem of counting ${\mathcal{M}}_U(x)$, the number of integers $n\le x$ such that $U_n=u^2+n{v}^2$ 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 González and the second author that ${\mathcal{M}}_F(x)\ll x(\log x{)}^{-0.06}$, with $\mathbf{F}$ the Fibonacci sequence.

19 pages