Fibonacci Sequence, Golden Ratio and Generalized Additive Sequences

In this article, we recall the Fibonacci sequence, the golden ratio, their properties and applications, and some early generalizations of the golden ratio. The Fibonacci sequence is a 2-sequence because it is generated by the sum of two previous terms. As a natural extension of this, we introduce several typical p-sequences where every term is the sum of p-previous terms given p initial values called seeds. In particular, we introduce the notion of 1-sequence. We then discuss generating functions and limiting ratio values of p-sequences. Furthermore, inspired by Fibonacci's rabbit pair problem, we consider a general problem whose particular cases lead to nontrivial additive sequences.