Lucas sequence

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Not to be confused with the Lucas numbers, which are one Lucas sequence.

In mathematics, a Lucas sequence is a particular generalisation of the Fibonacci numbers and Lucas numbers. Lucas sequences are named after French mathematician Edouard Lucas.

1 Recurrence relations
2 Algebraic relations
3 Other relations
4 Specific names
5 Applications
6 References

[edit] Recurrence relations

Given two integer parameters P and Q which satisfy

$P^2 - 4Q \neq 0$

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

$U_0(P,Q)=0 \,$

$U_1(P,Q)=1 \,$

$U_n(P,Q)=PU_{n-1}(P,Q)-QU_{n-2}(P,Q) \mbox{ for }n>1 \,$

and

$V_0(P,Q)=2 \,$

$V_1(P,Q)=P \,$

$V_n(P,Q)=PV_{n-1}(P,Q)-QV_{n-2}(P,Q) \mbox{ for }n>1 \,$

[edit] Algebraic relations

If the roots of the characteristic equation

$x^2 - Px + Q=0 \,$

are a and b with a > b or (a - b)/i > 0, then U(P,Q) and V(P,Q) can also be defined in terms of a and b by

$U_n(P,Q)= \frac{a^n-b^n}{a-b} = \frac{a^n-b^n}{ \sqrt{P^2-4Q}}$

$V_n(P,Q)=a^n+b^n \,$

from which we can derive the relations

$a^n = \frac{V_n + U_n \sqrt{P^2-4Q}}{2}$

$b^n = \frac{V_n - U_n \sqrt{P^2-4Q}}{2}$

(where the square root means its principal value).

Note that a and b are distinct because $P 2 - 4 Q$ is not 0.

[edit] Other relations

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

General	P=1, Q=-1
$U_n = \frac{V_{n+1} - Q V_{n-1}}{P^2-4Q}$	$U_n = \frac{V_{n+1} + V_{n-1}}{5}$
$V n = U n + 1 - Q U n - 1$	$V n = U n + 1 + U n - 1$
$U 2 n = U n V n$	$U 2 n = U n V n$
$V_{2n} = V_n^2 - 2Q^n$	$V_{2n} = V_n^2 - 2(-1)^n$
$U n + m = U n U m + 1 - Q U m U n - 1$	$U n + m = U n U m + 1 + U m U n - 1$
$V_{n+m} = V_n V_m - Q^m V_{n-m} \,$	$V_{n+m} = V_n V_m - (-1)^m V_{n-m} \,$