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.

[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) = P U n - 1 (P, Q) - Q U n - 2 (P, Q) for n > 1

and

V 0 (P, Q) = 2

V 1 (P, Q) = P

V n (P, Q) = P V n - 1 (P, Q) - Q V n - 2 (P, Q) for n > 1

If the roots of the characteristic equation

x 2 - P x + Q = 0

are a and b 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

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

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

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

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

$U_n = \frac{V_{n+1} - Q V_{n-1}}{P^2-4Q}$

V n = U n + 1 - Q U n - 1

U 2 n = U n V n

$V_{2n} = V_n^2 - 2Q^n$

U n + m = U n U m + 1 - Q U m U n - 1

V n + m = V n V m - Q m V n - m

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

V_n(1,−1) : Lucas numbers

U_n(2,−1) : Pell numbers

U_n(1,−2) : Jacobsthal numbers

Ribenboim, Paulo (2000). My Numbers, My Friends: Popular Lectures on Number Theory. Springer-Verlag New York Inc.. ISBN 0-387-98911-0.