The Fibonacci Numbers—Exposed

Among numerical sequences, the Fibonacci numbers Fn have achieved a kind of celebrity status. Indeed, Koshy gushingly refers to them as one of the “two shining stars in the vast array of integer sequences” [16, p. xi]. The second of Koshy’s “shining stars” is the Lucas numbers, a close relative of the Fibonacci numbers, about which we will say more below. The Fibonacci numbers are famous for possessing wonderful and amazing properties. Some are well known. For example, the sums and differences of Fibonacci numbers are Fibonacci numbers, and the ratios of Fibonacci numbers converge to the golden mean. Others are less familiar. Did you know that any four consecutive Fibonacci numbers can be combined to form a Pythagorean triple? Or how about this: The greatest common divisor of two Fibonacci numbers is another Fibonacci number. More precisely, the gcd of Fn and Fm is Fk , where k is the gcd of n and m. With such fabulous properties, it is no wonder that the Fibonacci numbers stand out as a kind of super sequence. But what if it is not such a special sequence after all? What if it is only a rather pedestrian sample from an entire race of super sequences? In this case, the home world is the planet of two term recurrences. As we shall show, its inhabitants are all just about as amazing as the Fibonacci sequence. The purpose of this paper is to demonstrate that many of the properties of the Fibonacci numbers can be stated and proved for a much more general class of sequences, namely, second-order recurrences. We shall begin by reviewing a selection of the properties that made Fibonacci numbers famous. Then there will be a survey of secondorder recurrences, as well as general tools for studying these recurrences. A number of the properties of the Fibonacci numbers will be seen to arise simply and naturally as the tools are presented. Finally, we will see that Fibonacci connections to Pythagorean triples and the gcd function also generalize in a natural way.