Lehmer sequence

In mathematics, a Lehmer sequence is a generalization of a Lucas sequence.^[1]

Algebraic relations

a + b = \sqrt{R}

a b = Q

under the following conditions:

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.

Lehmer numbers form a linear recurrence relation with

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

with initial values $U_{0} = 0, U_{1} = 1, U_{2} = 1, U_{3} = R - Q = a^{2} + a b + b^{2}$ . Similarly the companion sequence satisfies

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

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

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