On the Fibonacci and the Generalized Fibonacci Sequence

Fibonacci numbers and their sequence are found abundantly in nature. There is a close relation among the Golden, Fibonacci, and Lucas ratios. Such ratios are inherent to design, architecture, construction, and even to the beauty of different natural and manmade solid objects. Other variants, like the k-generalized Fibonacci sequence, the Fibonacci p-numbers, Lucas numbers, and the ordinary Fibonacci numbers, have some interesting properties and are based on recurrent relations. This article describes the different structures, mathematical beauty, and identities related to such sequences and numbers. Some important applications of Fibonacci sequences are also explored.