Generalized Fibonacci sequences and their properties

TL;DR

This paper explores properties of generalized Fibonacci sequences defined by multiple previous terms, providing new sum representations, deriving properties, and analyzing their 2-adic orders for various parameters.

Contribution

It introduces a novel sum-based expression for generalized Fibonacci numbers and analyzes their properties and 2-adic valuations.

Findings

Any $F_n(k)$ can be expressed as a sum of $B_n(k,j)$s.

Finite sum representations for $B_n(k,j)$ and $F_n(k)$ are derived.

The 2-adic order of these sequences is evaluated for various parameters.

Abstract

Let $F_n(k)$ be the generalized Fibonacci number defined by (with $F_i(k)$ abbreviated to $F_i$): $F_n = F_{n-1} + F_{n-2} + \dots + F_{n-k}$, for $n \geq k$, and the initial values $(F_0,F_1,...,F_{k-1})$. Let $B_n(k,j)$ be $F_n(k)$ with initial values given by $F_j = 1$ and, for $i<j$ and $j<i<k$, $F_i = 0$. This paper shows that any $F_n(k)$ can be expressed as the sum of $B_n(k,j)$s. This paper also expresses $B_n(k,j)$ and $F_n(k)$ as finite sums, derives some properties and evaluates their 2-adic order for a range of values of $k, j$ and $n$ and those of $B_n(3,j)$ and $B_n(4,j)$ for most values of $j$ and $n$.