Lucas sequence
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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
- <math>P^2 - 4Q > 0<math>
the Lucas sequences U(P,Q) and V(P,Q) are defined by the recurrence relations
- <math>U_0(P,Q)=0<math>
- <math>U_1(P,Q)=1<math>
- <math>U_n(P,Q)=PU_{n-1}(P,Q)-QU_{n-2}(P,Q) \mbox{ for }n>1<math>
and
- <math>V_0(P,Q)=2<math>
- <math>V_1(P,Q)=P<math>
- <math>V_n(P,Q)=PV_{n-1}(P,Q)-QV_{n-2}(P,Q) \mbox{ for }n>1<math>
Algebraic relations
If the roots of the characteristic equation
- <math>x^2 - Px + Q=0<math>
are a and b then U(P,Q) and V(P,Q) can also be defined in terms of a and b by
- <math>U_n(P,Q)= \frac{a^n-b^n}{a-b} = \frac{a^n-b^n}{ \sqrt{P^2-4Q}}<math>
- <math>V_n(P,Q)=a^n+b^n<math>
from which we can derive the relations
- <math>a^n = \frac{V_n + U_n \sqrt{P^2-4Q}}{2}<math>
- <math>b^n = \frac{V_n - U_n \sqrt{P^2-4Q}}{2}<math>
Other relations
The numbers in Lucas sequences satisfy relations that are analogues of the relations between Fibonacci numbers and Lucas numbers. For example :-
- <math>U_n = \frac{V_{n-1} + V_{n+1}}{P^2-4Q}<math>
- <math>V_n = U_{n-1} + U_{n+1}<math>
- <math>U_{2n} = U_n V_n<math>
- <math>V_{2n} = V_n^2 - 2Q^n<math>
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