Remarks On General Fibonacci Numbers

TL;DR

This paper explores the most generalized Fibonacci sequence, offering new proofs and properties, including divisibility conditions, infinite alternating bisquable sequences, and divisor bounds, expanding understanding of Fibonacci generalizations.

Contribution

It presents a new proof for the generalized Fibonacci sequence's properties, discusses divisibility restrictions, and introduces the concept of infinite alternating bisquable sequences.

Findings

Only the common general Fibonacci sequence can be divisible under certain restrictions.

Existence of infinite alternating bisquable Fibonacci sequences.

Provides a lower bound on the number of divisors of Fibonacci numbers.

Abstract

We dedicate this paper to investigate the most generalized form of Fibonacci Sequence, one of the most studied sections of the mathematical literature. One can notice that, we have discussed even a more general form of the conventional one. Although it seems the topic in the first section has already been covered before, but we present a different proof here. Later I found out that, the auxiliary theorem used in the first section was proven and even generalized further by F. T. Howard. Thanks to Curtis Cooper for pointing out the fact that this has already been studied and providing me with references. the At first, we prove that, only the common general Fibonacci Sequence can be a divisible sequence under some restrictions. In the latter part, we find some properties of the sequence, prove that there are infinite alternating bisquable Fibonacci sequence(defined later) and provide a lower bound on the number of divisors of Fibonacci numbers.