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

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 ${\displaystyle V_{n}(P,Q)}$.

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