= Lucas V-sequence
{c}
{title2=$V_n(P,Q)$}
= V-sequence
{c}
{synonym}
The <Lucas V-sequence> has $V_0(P,Q)=2$, $V_1(P,Q)=P$ and $V_{n+2}=PV_{n+1}-QV_n$. Thus its first terms are $2,P,P^2-2Q,P^3-3PQ$. For $(P,Q)=(1,-1)$ these are the Lucas numbers $2,1,3,4,7,\ldots$.
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 $V_n=2U_{n+1}-PU_n$ relates it to the <Lucas U-sequence> without requiring division.
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