Lehmer sequence

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

Algebraic relations
If a and b are complex numbers with


 * $$a + b = \sqrt{R}$$
 * $$ab = Q$$

under the following conditions:


 * Q and R are relatively prime nonzero integers
 * $$a/b$$ is not a root of unity.

Then, the corresponding Lehmer numbers are:


 * $$U_n(\sqrt{R},Q) = \frac{a^n-b^n}{a-b}$$

for n odd, and


 * $$U_n(\sqrt{R},Q) = \frac{a^n-b^n}{a^2-b^2}$$

for n even.

Their companion numbers are:


 * $$V_n(\sqrt{R},Q) = \frac{a^n+b^n}{a+b}$$

for n odd and


 * $$V_n(\sqrt{R},Q) = a^n+b^n$$

for n even.

Recurrence
Lehmer numbers form a linear recurrence relation with


 * $$U_n = (R-2Q)U_{n-2}-Q^2U_{n-4} = (a^2+b^2)U_{n-2}-a^2b^2U_{n-4}$$

with initial values $$U_0=0,\, U_1=1,\, U_2=1,\, U_3=R-Q=a^2+ab+b^2$$. Similarly the companion sequence satisfies


 * $$V_n = (R-2Q)V_{n-2}-Q^2V_{n-4} = (a^2+b^2)V_{n-2}-a^2b^2V_{n-4}$$

with initial values $$V_0=2,\, V_1=1,\, V_2=R-2Q=a^2+b^2,\, V_3=R-3Q=a^2-ab+b^2.$$