Concordance of certain 3-braids and Gauss diagrams

Let $\beta :={\sigma }_1{\sigma }_2^{-1}$ be a braid in $B_3$, where $B_3$ is the braid group on 3 strings and ${\sigma }_1,{\sigma }_2$ are the standard Artin generators. We use Gauss diagram formulas to show that for each natural number $n$ not divisible by $3$ the knot which is represented by the closure of the braid ${\beta }^n$ is algebraically slice if and only if $n$ is odd. As a consequence, we deduce some properties of Lucas numbers.