Lehmer sequence

In mathematics, a Lehmer sequence is a generalization of a Lucas sequence.[1]

Algebraic relations

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.

Recurrence

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

Reference

1. Weisstein, Eric W. "Lehmer Number". mathworld.wolfram.com. Retrieved 2020-08-11.