Editing Lucas sequence (section)

{{Short description|Certain constant-recursive integer sequences}}
{{distinguish|text=the sequence of [[Lucas number]]s, which is a particular Lucas sequence}}

In [[mathematics]], the '''Lucas sequences''' <math>U_n(P,Q)</math> and <math>V_n(P, Q)</math> are certain [[constant-recursive sequence|constant-recursive]] [[integer sequence]]s that satisfy the [[recurrence relation]]

: <math>x_n = P \cdot x_{n - 1} - Q \cdot x_{n - 2}</math>

where <math>P</math> and <math>Q</math> are fixed [[integer]]s. Any sequence satisfying this recurrence relation can be represented as a [[linear combination]] of the Lucas sequences <math>U_n(P, Q)</math> and <math>V_n(P, Q).</math>

More generally, Lucas sequences <math>U_n(P, Q)</math> and <math>V_n(P, Q)</math> represent sequences of [[polynomial]]s in <math>P</math> and <math>Q</math> with integer [[coefficient]]s.

Famous examples of Lucas sequences include the [[Fibonacci number]]s, [[Mersenne number]]s, [[Pell number]]s, [[Lucas number]]s, [[Jacobsthal number]]s, and a superset of [[Fermat number]]s (see below). Lucas sequences are named after the [[France|French]] mathematician [[Édouard Lucas]].