Lehmer sequence

In mathematics, a Lehmer sequence or is a generalization of a Lucas sequence or, 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 compared to the corresponding Lucas sequence. That is, when R = P² the Lehmer and Lucas sequences are related as:

Algebraic relations

If a and b are complex numbers with
under the following conditions:Q and R are relatively prime nonzero integers

Then, the corresponding Lehmer numbers are:
for n odd, and
for n even.
Their companion numbers are:
for n odd and
for n even.

Lehmer numbers form a linear recurrence relation with
with initial values. Similarly the companion sequence satisfies
with initial values
All Lucas sequence recurrences apply to Lehmer sequences if they are divided into cases for even and odd n and appropriate factors of are incorporated. For example,