Lucas sequence

In mathematics, the Lucas sequences ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$ are certain constant-recursive integer sequences that satisfy the recurrence relation

${\displaystyle x_{n}=P\cdot x_{n-1}-Q\cdot x_{n-2}}$

where ${\displaystyle P}$ and ${\displaystyle Q}$ are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q).}$

More generally, Lucas sequences ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$ represent sequences of polynomials in ${\displaystyle P}$ and ${\displaystyle Q}$ with integer coefficients.

Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers (see below). Lucas sequences are named after the French mathematician Édouard Lucas.