Counting terms Un of third order linear recurrences with Un= u2 + nv2

Abstract

Given a recurrent sequence U := {Un}n::a:O we consider the problem of counting Mu(x), the number of integers 11 ::: x such that U11 = u2 + nv2 for some integers u, v. We will show that Mu(x) « x(log x)-o.os for a large class of ternary sequences. Our method uses many ingredients from the proof of Alba Gonzalez and the second author [1] that MF(x) « x(logx)-0· 06 , with F the Fibonacci sequence.