Additive Sequences, Sums, Golden Ratios and Determinantal Identities

Abstract

The Fibonacci sequence is a series of positive integers in which, starting from $0$ and $1$, every number is the sum of two previous numbers, and the limiting ratio of any two consecutive numbers of this sequence is called the golden ratio. The Fibonacci numbers and the golden ratio are two significant concepts that keep appearing everywhere. In this article, we investigate the following issues: (i) 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, $f_{n+2} = f_{n+1} + f_{n}$. 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. (ii) We obtain closed expressions for odd and even sums, sum of the first $n$ numbers, and the sum of squares of the first $n$ numbers of the "exponent" $p$-sequence whose seeds are $(0,1,\cdots,p-1)$. (iii) We investigate the $p$-golden ratio of $p$-sequences, express a positive integer power of the $p$-golden ratio as a polynomial of degree $p-1$, and obtain values of golden angles for different $p$-golden ratios. We also consider further generalizations of the golden ratio. (iv) We establish a family of determinantal identities of which the Cassini's identity is a particular case.