In mathematics, a Lehmer sequence
n
{\displaystyle U_{n}({\sqrt {R}},Q)}
n
,
)
is a generalization of a Lucas sequence
, 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: If a and b are complex numbers with under the following conditions: 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 √R are incorporated.
This number theory-related article is a stub.
You can help Wikipedia by expanding it.