Concordance of certain 3-braids and Gauss diagrams

Let β:=σ1σ2−1 be a braid in B3B3, where B3B3 is the braid group on 3 strings and σ1σ1, σ2σ2 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 βnβn is algebraically slice if and only if n is odd. As a consequence, we deduce some properties of Lucas numbers.