The term Fibonacci sequence is also applied more generally to any function g where g(n + 2) = g(n) + g(n + 1). These functions are precisely those with the form g(n) = aF(n) + bF(n - 1) for some a, b, so the Fibonacci sequences form a vector space with the functions F(n) and F(n - 1) as a basis.
As was pointed out by Johannes Kepler, the growth rate of the Fibonacci numbers, that is, F(n + 1) / F(n), converges to the golden mean, denoted φ. This is the positive root of the quadratic x2 - x - 1, so φ2 = φ + 1. If we multiply both sides by φn, we get φn+2 = φn+1 + φn, so the function φn is a Fibonacci sequence. The negative root of the quadratic, 1 - φ, can be shown to have the same properties, so the two functions φn and (1-φ)n form another basis for the space.
This gives us a formula for the normal Fibonacci sequence:
F(n) = φn / √5 - (1-φ)n / √5
As n goes to infinity, the second term converges to zero, so the Fibonacci numbers approach the exponential φn / √5, hence their convergent ratios. In fact the second term starts out small enough that the Fibonacci numbers can be obtained from the first term alone, by rounding to the nearest integer. Matiyasevich was able to show that they can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem.
A generalization of the Fibonacci sequence are the [Lucas sequence]?s. One kind can be defined thus:
L(0) = 0 L(1) = 1 L(n+2) = PL(n+1) + QL(n)
where the normal Fibbonaci sequence is the special case where P = Q = 1. Another kind of Lucas Sequence begins with L(0) = 2, L(1) = P. Such sequences have applications in number theory and primality proving.