Additive Sequences, Sums, Golden Ratios and Determinantal Identities
Asutosh Kumar

TL;DR
This paper explores generalized additive sequences, their sums, ratios, and identities, extending Fibonacci concepts to p-sequences and introducing new mathematical properties and identities related to these sequences.
Contribution
It introduces p-sequences as generalizations of Fibonacci, derives formulas for their sums, explores p-golden ratios, and establishes new determinantal identities.
Findings
Closed-form expressions for sums of p-sequences.
Introduction of p-golden ratios and their properties.
Family of determinantal identities generalizing Cassini's identity.
Abstract
The Fibonacci sequence is a series of positive integers in which, starting from and , 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 -sequence because it is generated by the sum of two previous terms, . As a natural extension of this, we introduce several typical -sequences where every term is the sum of previous terms given initial values called "seeds". In particular, we introduce the notion of -sequence. We then…
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