# Lucas sequence

In mathematics a Lucas sequence is a particular generalisation of the Fibonacci numbers and Lucas numbers. Lucas sequences were first studied by French mathematician Edouard Lucas.

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## Recurrence relations

Given two integer parameters P and Q which satisfy

[itex]P^2 - 4Q > 0[itex]

the Lucas sequences U(P,Q) and V(P,Q) are defined by the recurrence relations

[itex]U_0(P,Q)=0[itex]
[itex]U_1(P,Q)=1[itex]
[itex]U_n(P,Q)=PU_{n-1}(P,Q)-QU_{n-2}(P,Q) \mbox{ for }n>1[itex]

and

[itex]V_0(P,Q)=2[itex]
[itex]V_1(P,Q)=P[itex]
[itex]V_n(P,Q)=PV_{n-1}(P,Q)-QV_{n-2}(P,Q) \mbox{ for }n>1[itex]

## Algebraic relations

If the roots of the characteristic equation

[itex]x^2 - Px + Q=0[itex]

are a and b then U(P,Q) and V(P,Q) can also be defined in terms of a and b by

[itex]U_n(P,Q)= \frac{a^n-b^n}{a-b} = \frac{a^n-b^n}{ \sqrt{P^2-4Q}}[itex]
[itex]V_n(P,Q)=a^n+b^n[itex]

from which we can derive the relations

[itex]a^n = \frac{V_n + U_n \sqrt{P^2-4Q}}{2}[itex]
[itex]b^n = \frac{V_n - U_n \sqrt{P^2-4Q}}{2}[itex]

## Other relations

The numbers in Lucas sequences satisfy relations that are analogues of the relations between Fibonacci numbers and Lucas numbers. For example :-

[itex]U_n = \frac{V_{n-1} + V_{n+1}}{P^2-4Q}[itex]
[itex]V_n = U_{n-1} + U_{n+1}[itex]
[itex]U_{2n} = U_n V_n[itex]
[itex]V_{2n} = V_n^2 - 2Q^n[itex]

## Specific names

The Lucas sequences for some values of P and Q have specific names :-

Un(1,−1) : Fibonacci numbers
Vn(1,−1) : Lucas numbers
Un(2,−1) : Pell numbers
Un(1,−2) : Jacobsthal numbersde:Lucas-Folge

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