The Lucas sequence describes a pattern of integers that follows a specific recurrence relation. The number series that falls into the category of a Lucas sequence conforms to the formula X(n) = PX(n−1) - QX(n−2), where P and Q are fixed integers. The Fibonacci number series is a famous example of a Lucas sequence.
The Lucas sequence is usually denoted by the terms Un(P,Q) and Vn(P,Q). Any integer sequence that follows the aforementioned recurrence relation may be signified by a linear combination of the Lucas sequence. The term Un(P,Q) may be defined as the difference of the nth power of the roots divided by the difference of the said roots. The term Vn(P,Q) is the sum of the nth power of the roots. To determine the terms of the Lucas sequence, the value of n should start with zero to infinity. Hence, if n is equal to zero, the values of Un(P,Q) and Vn(P,Q) are zero and two, respectively. The sequence continues as the numerical substitution of n progresses.
Although they may appear the same, Lucas numbers are different from the Lucas sequence. Lucas numbers are composed of a series of integers that follow a similar pattern as the Fibonacci sequence but at a slight variation. The Lucas numbers follow the formula L(n+1) = L(n) + L(n-1), with the first two numbers in the sequence being 1 and 3.
The Lucas sequence was named in honor of Édouard Lucas, a French mathematician.