Lucas sequence

In mathematics, the Lucas sequences U n ( P , Q ) {displaystyle U_{n}(P,Q)} and V n ( P , Q ) {displaystyle V_{n}(P,Q)} are certain constant-recursive integer sequences that satisfy the recurrence relation In mathematics, the Lucas sequences U n ( P , Q ) {displaystyle U_{n}(P,Q)} and V n ( P , Q ) {displaystyle V_{n}(P,Q)} are certain constant-recursive integer sequences that satisfy the recurrence relation where P {displaystyle P} and Q {displaystyle Q} are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U n ( P , Q ) {displaystyle U_{n}(P,Q)} and V n ( P , Q ) {displaystyle V_{n}(P,Q)} . More generally, Lucas sequences U n ( P , Q ) {displaystyle U_{n}(P,Q)} and V n ( P , Q ) {displaystyle V_{n}(P,Q)} represent sequences of polynomials in P {displaystyle P} and Q {displaystyle Q} with integer coefficients. Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers. Lucas sequences are named after the French mathematician Édouard Lucas. Given two integer parameters P and Q, the Lucas sequences of the first kind Un(P,Q) and of the second kind Vn(P,Q) are defined by the recurrence relations: