 Lucas V-sequence

The Lucas V-sequence has $V_0(P,Q)=2$, $V_1(P,Q)=P$ and $V_{n+2}=P{V}_{n+1}-Q{V}_n$. Thus its first terms are $2,P,P^2-2Q,P^3-3PQ$. For $(P,Q)=(1,-1)$ these are the Lucas numbers $2,1,3,4,7,\dots$.

For roots $\alpha ,\beta$ of $t^2-Pt+Q$, one has $V_n={\alpha }^n+{\beta }^n$, including the repeated-root case. This gives $V_{2n}=V_n^2-2Q^n$ immediately. The identity $V_n=2U_{n+1}-P{U}_n$ relates it to the Lucas U-sequence without requiring division.