On a New Formula for Fibonacci’s Family m-step Numbers and Some Applications

Abstract

In this work, we obtain a new formula for Fibonacci’s family m-step sequences. We use our formula to find the nth term with less time complexity than the matrix multiplication method. Then, we extend our results for all linear homogeneous recurrence m-step relations with constant coefficients by using the last few terms of its corresponding Fibonacci’s family m-step sequence. As a computational number theory application, we develop a method to estimate the square roots.

Keywords

• Fibonacci’s <i>m</i>-step numbers
• time complexity
• linear homogeneous recurrence relations