Lucas sequence

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

Recurrence relations

Given two integer parameters P and Q which satisfy

$P^{2}-4Q>0$

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

$V_{0}(P,Q)=2$

$V_{1}(P,Q)=P$

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

Algebraic relations

If the roots of the characteristic equation

$x^{2}-Px+Q=0$

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

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

$b^{n}={\frac {V_{n}-U_{n}{\sqrt {P^{2}-4Q}}}{2}}$

Other relations

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

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

$V_{n}=U_{n-1}+U_{n+1}$

$U_{2n}=U_{n}V_{n}$

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

Specific names

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

U_n(1,−1) : Fibonacci numbers

V_n(1,−1) : Lucas numbers

U_n(2,−1) : Pell numbers

U_n(1,−2) : Jacobsthal numbers