In this paper we introduce generalized Mersenne numbers. We shall present some of their interpretations and matrix generators which are very useful for determining identities.
@article{bwmeta1.element.ojs-doi-10_17951_a_2018_72_1_69-76,
author = {Pawe\l\ Ochalik and Andrzej W\l och},
title = {On generalized Mersenne numbers, their interpretations and matrix generators},
journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
volume = {72},
year = {2018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2018_72_1_69-76}
}
Paweł Ochalik; Andrzej Włoch. On generalized Mersenne numbers, their interpretations and matrix generators. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 72 (2018) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2018_72_1_69-76/
Berge, C., Principles of Combinatorics, Academic Press, New York-London, 1971.
Civciv, H., Turkman, R., On the (s,t)-Fibonacci and Fibonacci Matrix Sequences, Ars Combin. 87 (2008), 161-173.
Ercolano, J., Matrix generator of Pell sequences, Fibonacci Quart. 17 (1) (1979), 71-77.
Kaygisiz, K., Sahin, A., Determinant and permanent of the Hessenberg matrix and Fibonacci type numbers, Gen. Math. Notes 9 (2) (2012), 32-41.
Kilic, E., On the usual Fibonacci and generalized order k-Pell sequences by Hessenberg matrices, Ars Combin. 94 (2010), 161-174.
Kilic, E., Stanica, P., A matrix approach for general higher order linear recurrence, Bull. Malays. Math. Sci. Soc. (2) 34 (1) (2011), 51-67.
Kilic, E., Tasci, D., On the generalized Fibonacci and Pell sequences by Hessenberg matrices, Ars Combin. 94 (2010) 161-174.
Sergeer, A. S., Generalized Mersenne matrices and Balonin’s conjecture, Automatic Control and Computer Sciences 48 (4) (2014), 214-220.
Solinas, J., Generalized Mersenne Numbers, Technical report CORR-39, Dept. of C. & O., University of Waterloo, 1999.
Available from http://www.carc.math.uwaterloo.ca
Włoch, A., Wołowiec-Musiał, M., Generalized Pell numbers and some relations with Fibonacci numbers, Ars Combin. 109 (2013), 391-403.
Zheng, Y., Shon, S., Exact inverse matrices of Fermat and Mersenne circulant matrix, Abstr. Appl. Anal. 2015 (2015), Article 760823, 10 pp.