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.
Highlights
In modern science, extensive work has been done in the area of recurrence relations and their applications.In [6,7], the authors developed a transformation method of Tribonacci sequence and Tetranacci sequence to find the nth term of any Tribonacci-Like sequence and Tetranacci-Like sequence, respectively
The time complexity of our formula is of order m2 log n times the time of multiplying two n-digit integers
As a computational number theory application, we develop a method to estimate the square root
Summary
In modern science, extensive work has been done in the area of recurrence relations and their applications (see, e.g., [1,2,3,4,5]). In [9], the authors used matrix multiplication to find the nth term of Fibonacci’s family m-step sequences. The time complexity of their result is of order m3 log n times the time of multiplying two n-digit integers. The time complexity of our formula is of order m2 log n times the time of multiplying two n-digit integers. Time complexity shows how fast the algorithm converges to the solution
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