A linear homogeneous recurrence relation of degree 2 with constant coefficients is an equation of the form
where c1 and c2 are real numbers with c2 ≠ 0. It is not difficult to show (see [Ro07]) that if the equation r2− c1r − c2= 0 has two distinct roots r1and r2, then the sequence {an}is a solution of the linear homogeneous recurrence relation an = c1an−1+ c2an−2 if and only if for n = 0, 1, 2, …, where C1 and C2 are constants. The values of these constants can be found using the two initial terms of the sequence.
Find an explicit formula for the Lucas numbers using the technique of Exercise
Exercise
Use the generating function where fk is the kth Fibonacci number to find an explicit formula for fk, proving Theorem 1.7. (Hint: Use the fact that fk = fk−1 + fk−2 for k = 2, 3, ... to show that G(x) − xG(x) − x2G(x) = x. Solve this to show that G(x) = x/( 1 − x − x2) and then write G(x) in terms of partial fractions, as is done in calculus.) (See [Ro07] for information on using generating functions.)
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