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.^[1]

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:

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

If a and b are complex numbers with

${\displaystyle a+b={\sqrt {R}}}$

${\displaystyle ab=Q}$

under the following conditions:

• Q and R are relatively prime nonzero integers
• ${\displaystyle a/b}$ is not a root of unity.

Then, the corresponding Lehmer numbers are:

${\displaystyle U_{n}({\sqrt {R}},Q)={\frac {a^{n}-b^{n}}{a-b}}}$

for n odd, and

${\displaystyle U_{n}({\sqrt {R}},Q)={\frac {a^{n}-b^{n}}{a^{2}-b^{2}}}}$

for n even.

Their companion numbers are:

${\displaystyle V_{n}({\sqrt {R}},Q)={\frac {a^{n}+b^{n}}{a+b}}}$

for n odd and

${\displaystyle V_{n}({\sqrt {R}},Q)=a^{n}+b^{n}}$

for n even.

Lehmer numbers form a linear recurrence relation with

${\displaystyle U_{n}=(R-2Q)U_{n-2}-Q^{2}U_{n-4}=(a^{2}+b^{2})U_{n-2}-a^{2}b^{2}U_{n-4}}$

with initial values ${\displaystyle U_{0}=0,\,U_{1}=1,\,U_{2}=1,\,U_{3}=R-Q=a^{2}+ab+b^{2}}$. Similarly the companion sequence satisfies

${\displaystyle V_{n}=(R-2Q)V_{n-2}-Q^{2}V_{n-4}=(a^{2}+b^{2})V_{n-2}-a^{2}b^{2}V_{n-4}}$

with initial values ${\displaystyle V_{0}=2,\,V_{1}=1,\,V_{2}=R-2Q=a^{2}+b^{2},\,V_{3}=R-3Q=a^{2}-ab+b^{2}.}$

All Lucas sequence recurrences apply to Lehmer sequences if they are divided into cases for even and odd n and appropriate factors of √R are incorporated. For example,

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