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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
![{\displaystyle P^{2}-4Q>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/847dce81a6151029e1cc4092aa9b9fe25ee2cfca)
the Lucas sequences U(P,Q) and V(P,Q) are defined by the recurrence relations
![{\displaystyle U_{0}(P,Q)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/533aecf96cd95bde04cf9090db00a4a1fe969456)
![{\displaystyle U_{1}(P,Q)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45876a4a48e4702717cd9664f915ca95a08db92b)
![{\displaystyle U_{n}(P,Q)=PU_{n-1}(P,Q)-QU_{n-2}(P,Q){\mbox{ for }}n>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/347879aa709bfd4f83dec50e73f5d3a8faa01da3)
and
![{\displaystyle V_{0}(P,Q)=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/063fd818f43d67af283d935fbd54983ee98ee8e7)
![{\displaystyle V_{1}(P,Q)=P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e45683f65eae1f0c21d61bf0722862a3d0f0ec6)
![{\displaystyle V_{n}(P,Q)=PV_{n-1}(P,Q)-QV_{n-2}(P,Q){\mbox{ for }}n>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c04271e1cddf6403702f5eadf42cb01909c99475)
Algebraic relations
If the roots of the characteristic equation
![{\displaystyle x^{2}-Px+Q=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/331d7569dafe3cf73077366a07a78dc4f6b4ec3e)
are a and b then U(P,Q) and V(P,Q) can also be defined in terms of a and b by
![{\displaystyle U_{n}(P,Q)={\frac {a^{n}-b^{n}}{a-b}}={\frac {a^{n}-b^{n}}{\sqrt {P^{2}-4Q}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/209fcd71157aeb3bf30b3a84cea6dde5ef0b5195)
![{\displaystyle V_{n}(P,Q)=a^{n}+b^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e71629a1114d22ad063067a193fbde64a0b5fb)
from which we can derive the relations
![{\displaystyle a^{n}={\frac {V_{n}+U_{n}{\sqrt {P^{2}-4Q}}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bdc9d2a1f17c3bc0239cacdb5bb07d4190fcd4d)
![{\displaystyle b^{n}={\frac {V_{n}-U_{n}{\sqrt {P^{2}-4Q}}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4ab289b4051389e60dec84a7dfc46439a59297)
Other relations
The numbers in Lucas sequences satisfy relations that are analogues of the relations between Fibonacci numbers and Lucas numbers. For example :-
![{\displaystyle U_{n}={\frac {V_{n-1}+V_{n+1}}{P^{2}-4Q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcfeac068fdb97bcff03b1eda93b7a79dc45d9ad)
![{\displaystyle V_{n}=U_{n-1}+U_{n+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dac85f8f52a9d82265d6c54538b916083eea116c)
![{\displaystyle U_{2n}=U_{n}V_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc060eff26eab63082b935a4f59c6784d738755b)
![{\displaystyle V_{2n}=V_{n}^{2}-2Q^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf95e7fab570998290688d32240dfe2517721e17)
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 numbers