Lehmer sequence

In mathematics, a Lehmer sequence ${\displaystyle U_{n}({\sqrt {R}},Q)}$ or ${\displaystyle V_{n}({\sqrt {R}},Q)}$ is a generalization of a Lucas sequence ${\displaystyle U_{n}(P,Q)}$ or ${\displaystyle V_{n}(P,Q)}$, allowing the square root of an integer R in place of the integer P.

To ensure that the value is always an integer, every other term of a Lehmer sequence is divided by √R compared to the corresponding Lucas sequence. That is, when R = P² the Lehmer and Lucas sequences are related as:

${\displaystyle {\begin{aligned}P\,U_{2n}({\sqrt {P^{2}}},Q)&=U_{2n}(P,Q)&U_{2n+1}({\sqrt {P^{2}}},Q)&=U_{2n+1}(P,Q)\\V_{2n}({\sqrt {P^{2}}},Q)&=V_{2n}(P,Q)&P\,V_{2n+1}({\sqrt {P^{2}}},Q)&=V_{2n+1}(P,Q)\end{aligned}}}$