A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions of n + 1 into parts greater than one. Some commentary about the history of partitions and compositions is provided.

The Binet formula gives a natural way for Fibonacci numbers to be viewed as a function of a complex variable. We experimentally study the complex dynamics of the Fibonacci numbers viewed in that manner. Attracting and repelling fixed points are related to the filled Julia set and to regions of escape time images with fascinating behavior. Introduction The Fibonacci numbers are traditionally des...

The extended Fibonacci sequence of numbers and polynomials is introduced and studied. The generating function, recurrence relations, an expansion in terms of multinomial coefficients, and several properties of the extended Fibonacci numbers and polynomials are obtained. Interesting relations between them and probability problems which take into account lengths of success and failure runs are al...

Webb & Parberry proved in 1969 a startling trigonometric identity involving Fibonacci numbers. This identity has remained isolated up to now, despite the amount of work on related polynomials. We provide a wide generalization of this identity together with what we believe (and hope!) to be its proper understanding.

A Diophantine m-tuple is a set of m positive integers with the property that product of any two of its distinct elements is one less then a square. In this survey we describe some problems and results concerning Diophantine m-tuples and their connections with Fibonacci numbers.

The Fibonacci cube [6] is a new class of graphs that are inspired by the famous numbers. Because of the rich properties of the Fibonacci numbers [1], the graph also shows interesting properties. For a graph with AT nodes, it is known [6] that the diameter, the edge connectivity, and the node connectivity of the Fibonacci cube are in the order of 0(log N), which are similar to the Boolean cube (...

The Fibonacci sequence {Fn} is defined by the recurrence relation Fn = Fn−1+ Fn−2, for n ≥ 2 with F0 = 0 and F1 = 1. The Lucas sequence {Ln} , considered as a companion to Fibonacci sequence, is defined recursively by Ln = Ln−1 + Ln−2, for n ≥ 2 with L0 = 2 and L1 = 1. It is well known that F−n = (−1)Fn and L−n = (−1)Ln, for every n ∈ Z. For more detailed information see [9],[10]. This paper pr...