Lucas U-sequence ($U_n(P,Q)$)

The Lucas U-sequence has $U_0(P,Q)=0$, $U_1(P,Q)=1$ and $U_{n+2}=P{U}_{n+1}-Q{U}_n$. Its first terms are $0,1,P,P^2-Q,P^3-2PQ$. At $(P,Q)=(1,-1)$ it is the Fibonacci sequence.

If the roots $\alpha ,\beta$ of its characteristic polynomial of a linear recurrence are distinct, $U_n=({\alpha }^n-{\beta }^n)/(\alpha -\beta )$. This identity follows by checking the two initial conditions and the linear recurrence relation. At a repeated root $\alpha =\beta$, the corresponding formula is $U_n=n{\alpha }^{n-1}$ for $n\ge 1$.