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.[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:

Algebraic relations
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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. For example,

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