On generalized Mersenne numbers, their interpretations and matrix generators. (English) Zbl 1441.11027

Ochalik, Paweł; Włoch, Andrzej

Ann. Univ. Mariae Curie-Skłodowska, Sect. A 72, No. 1, 69-76 (2018).

The numbers of the form \(M_n=2^n-1\) are known as Mersenne numbers due to Marin Mersenne who studied these numbers in the 17th century. The primes of the form \(M_n\) are called Mersenne primes. And if \(M_n\) is prime then \(n\) must be prime. Also the \(n\)-th Mersenne number is the optimal solution of the Hanoi tower problem where \(n\) is the number of discs.
In the paper under review the authors introduce a generalization of Mersenne numbers. For a fixed positive integer \(k\ge 3\) the generalized Mersenne numbers are defined by the second order linear recurrence relation;
\[ M(k,n) = kM(k,n-1)-(k-1)M(k,n-2) \]
where \(M(k,0)=0\) and \(M(k,1)=1\). It can be seen from the definition that if \(k=3\) then \(M(3,n)=M_n\). First a Binet-type formula for the generalized Mersenne numbers is obtained by solving the linear recurrence. After that the authors give some important identities.
In section 2, the matrix generator for generalized Mersenne numbers is considered as;
\[\begin{bmatrix} M(k,n+2) & M(k,n+1) \\ M(k,n+1) & M(k,n)\end{bmatrix} = \begin{bmatrix} k & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} k & 1 \\ -(k-1) & 0\end{bmatrix}^n \]
Also some basic results are obtained.
In the same section the authors mention the lower Hessenberg matrix of size \(n\). They prove that the determinant of the \(n\)-th Hessenberg matrix is equal to \(M(k,n+1)\).
Combinatorial interpretations of the generalized Mersenne numbers are given in section 3.

Reviewer: Bilal Demir (Balıkesir)

MSC:

11B37	Recurrences
15B36	Matrices of integers

Keywords:

Mersenne numbers; Fibonacci numbers; matrix generators

References:

[1]	Berge, C., Principles of Combinatorics, Academic Press, New York-London, 1971 · Zbl 0227.05002
[2]	Civciv, H., Turkman, R., On the (s,t)-Fibonacci and Fibonacci Matrix Sequences, Ars Combin. 87 (2008), 161-173 · Zbl 1224.11015
[3]	Ercolano, J., Matrix generator of Pell sequences, Fibonacci Quart. 17 (1) (1979), 71-77 · Zbl 0399.10011
[4]	Kaygisiz, K., Sahin, A., Determinant and permanent of the Hessenberg matrix and Fibonacci type numbers, Gen. Math. Notes 9 (2) (2012), 32-41
[5]	Kilic, E., On the usual Fibonacci and generalized order k-Pell sequences by Hessenberg matrices, Ars Combin. 94 (2010), 161-174 · Zbl 1240.11036
[6]	Kilic, E., Stanica, P., A matrix approach for general higher order linear recurrence, Bull. Malays. Math. Sci. Soc. (2) 34 (1) (2011), 51-67 · Zbl 1246.11027
[7]	Kilic, E., Tasci, D., On the generalized Fibonacci and Pell sequences by Hessenberg matrices, Ars Combin. 94 (2010) 161-174 · Zbl 1240.11036
[8]	Sergeer, A. S., Generalized Mersenne matrices and Balonin’s conjecture, Automatic Control and Computer Sciences 48 (4) (2014), 214-220
[9]	Solinas, J., Generalized Mersenne Numbers, Technical report CORR-39, Dept. of C. & O., University of Waterloo, 1999
[10]	Available from http://www.carc.math.uwaterloo.ca
[11]	Włoch, A., Wołowiec-Musiał, M., Generalized Pell numbers and some relations with Fibonacci numbers, Ars Combin. 109 (2013), 391-403 · Zbl 1289.11013
[12]	Zheng, Y., Shon, S., Exact inverse matrices of Fermat and Mersenne circulant matrix, Abstr. Appl. Anal. 2015 (2015), Article 760823, 10 · Zbl 1383.15029