**5. Conclusions**

Mersenne numbers are remarkably wide-spread in many diverse areas of the mathematical, biological, physical, chemical, engineering, and statistical sciences. In this paper, we present explicit formulas for the determinants and inverses of periodic tridiagonal Toeplitz matrices with perturbed corners. The representation of the determinant in the form of products of the Mersenne numbers and some initial values from matrix transformations and Schur complement. For the inverse, our main approaches include the use of matrix decomposition with the Sherman-Morrison-Woodbury formula. Especially, the inverse is just determined by six initial values. To test our method's effectiveness, we propose two algorithms for finding the determinant and inverse of periodic tridiagonal Toeplitz matrices with perturbed corners and compare the total number of operations for the two basic quantities between different algorithms. After comparison, we draw a conclusion that our algorithms are superior to LU decomposition to some extent.

**Author Contributions:** Conceptualization, methodology, funding acquisition, Z.J. and Y.Z.; investigation, resources, formal analysis, software, Y.Z. and Y.W.; writing—original draft preparation, Y.W.; writing—review and editing, supervision, visualization, S.S.

**Funding:** The research was funded by Natural Science Foundation of Shandong Province (Grant No. ZR2016AM14), National Natural Science Foundation of China (Grant No.11671187) and the PhD Research Foundation of Linyi University (Grant No.LYDX2018BS067).

**Acknowledgments:** The authors are grateful to the anonymous referees for their useful suggestions which improve the contents of this article.

**Conflicts of Interest:** The authors declare no conflict of interest.
