Formula for the $n$th $k$-Generalized Fibonacci-like Number

TL;DR

This paper derives a general explicit formula for the nth term of the k-generalized Fibonacci-like sequence, extending known formulas for Fibonacci, Tribonacci, and Tetrabonacci sequences.

Contribution

It provides a new explicit formula for the k-generalized Fibonacci-like sequence, generalizing previous specific cases and using pattern observation and existing formulas.

Findings

Derived a formula for the nth term of the k-generalized Fibonacci-like sequence.

Connected the formula to known sequences like Fibonacci, Tribonacci, and Tetrabonacci.

Validated the formula through pattern analysis and existing sequence formulas.

Abstract

In this paper we provided a formula for the $n$th term of the $k$-generalized Fibonacci-like sequence, a generalization of the well-known Fibonacci sequence, having $k$ arbitrary initial terms, where the succeeding terms are obtained by adding its previous $k$ terms. The formula for the $n$th term of the $k$-generalized Fibonacci-like sequence was obtained by observing patterns in the derived formula for the nth term of the Fibonacci-like, Tribonacci-like, and Tetrabonacci-like sequence. The formula for the $k$-generalized Fibonacci sequence was also derived and was used in the process of proving the main result of this paper.