: Mathematics: Lucas number

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Lucas number

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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Lucas_sequence)

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