Lehmer number

In mathematics, a Lehmer number is a generalization of a Lucas sequence.

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

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

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