**Volume 23** ▶ Number 1 ▷ Number 2 ▷ Number 3

**On the extensibility of the D(4)-triple {***k*–2, *k*+2, 4*k*} over Gaussian integers

*Original research paper. Pages 1—26*

Abdelmejid Bayad, Appolinaire Dossavi-Yovo, Alan Filipin, and Alain Togbé

Full paper (PDF, 246 Kb) | Abstract

In this paper, we prove that if {

*k* – 2,

*k* + 2, 4

*k*}, where k ∈

,

*k* ≠ 0, ± 2, is a D(4)-quadruple in the ring of Gaussian integers, then

*d* = 4

*k*^{3 } − 4

*k*.

**All associated Stirling numbers are arithmetical triangles**

*Original research paper. Pages 27—34*

Khaled Ben Letaïef

Full paper (PDF, 176 Kb) | Abstract

Associated Stirling numbers of first and second kind are usually found in the literature in various forms of stairs depending on their order *r*. Yet, it is shown in this note that all of these numbers can be arranged, through a linear transformation, in the same arithmetical triangle structure as the “Pascal’s triangle”.

**A short proof of a concrete sum**

*Original research paper. Pages 35—37*

Samuel G. Moreno and Esther M. García-Caballero

Full paper (PDF, 133 Kb) | Abstract

We give an alternative proof of a formula that generalizes Hermite’s identity. Instead involving modular arithmetic, our short proof relies on the Fourier-type expansion for the floor function and on a trigonometric formula

**On ***k*-balancing numbers

*Original research paper. Pages 38—52*

Arzu Özkoç and Ahmet Tekcan

Full paper (PDF, 205 Kb) | Abstract

In this work, we consider some algebraic properties of *k-*balancing numbers. We deduce some formulas for the greatest common divisor of *k*-balancing numbers, divisibility properties of *k*-balancing numbers, sums of *k*-balancing numbers and simple continued fraction expansion of *k*-balancing numbers.

**The abundancy index of divisors of odd perfect numbers – Part III**

*Original research paper. Pages 53—59*

Jose Arnaldo B. Dris

Full paper (PDF, 139 Kb) | Abstract

Dris gave numerical bounds for the sum of the abundancy indices of *q*^{k} and *n*^{2}, where *q*^{k}*n*^{2} is an odd perfect number, in his master’s thesis. In this note, we show that improving the limits for this sum is equivalent to obtaining nontrivial bounds for the Euler prime *q*.

**On a Pillai’s Conjecture and gaps between consecutive primes**

*Original research paper. Pages 60—72*

Rafael Jakimczuk

Full paper (PDF, 199 Kb) | Abstract

In this article we show that the following Pillai’s conjecture

*p*_{n} is the

*n*-th prime number)

can be established in terms of gaps between consecutive primes. We also study general sequences that have this property. We call these sequences Pillaisequences. We prove that the sequence of perfect powers is a Pillai-sequence.

**On quasiperfect numbers**

*Original research paper. Pages 73—78*

V. Siva Rama Prasad and C. Sunitha

Full paper (PDF, 158 Kb) | Abstract

A natural number *N *is said to be quasiperfect if σ(N) = 2*N* + 1 where *σ*(*N*) is the sum of the positive divisors of *N*. No quasiperfect number is known. If a quasiperfect number N exists and if ω(*N*) is the number of distinct prime factors of *N* then G. L. Cohen has proved ω(*N*) ≥ 7 while H. L. Abbott et. al have shown ω(*N*) ≥ 10 if (*N*, 15) = 1. In this paper we first prove that every quasiperfect numbers *N* has an odd number of special factors (see definition 2.3 below) and use it to show that ω(*N*) ≥ 15 if (*N*, 15) = 1 which refines the result of Abbott et.al. Also we provide an alternate proof of Cohen’s result when (*N*, 15) = 5.

**A Wilf class composed of 19 symmetry classes of quadruples of 4-letter patterns**

*Original research paper. Pages 79—99*

Talha Arıkan, Emrah Kılıç and Toufik Mansour

Full paper (PDF, 240 Kb) | Abstract

In this paper, we make a contribution to the enumeration of permutations avoiding a quadruples of 4-letter patterns by establishing a Wilf class composed of 19 symmetry classes.

**Pell–Padovan-circulant sequences and their applications**

*Original research paper. Pages 100—114*

Ömür Deveci and Anthony G. Shannon

Full paper (PDF, 315 Kb) | Abstract

This paper develops properties of recurrence sequences defined from circulant matrices obtained from the characteristic polynomial of the Pell-Padovan sequence. The study of these sequences modulo m yields cyclic groups and semigroups from the generating matrices. Finally, we obtain the lengths of the periods of the extended sequences in the extended triangle groups *E*(2, *n*, 2), *E*(2, 2, *n*) and E(*n*, 2, 2) for *n* ≥ 3 as applications of the results obtained.

**On two new two-dimensional extensions of the Fibonacci sequence**

*Original research paper. Pages 115—122*

Krassimir T. Atanassov

Full paper (PDF, 133 Kb) | Abstract

Two new two-dimensional extensions of the Fibonacci sequence are introduced. Explicit formulas for their *n*-th members are given.

**Non-split domination subdivision critical graphs**

*Original research paper. Pages 123—132*

Girish V. R. and P. Usha

Full paper (PDF, 154 Kb) | Abstract

A set of vertices *S *is said to dominate the graph *G* if for each *v* ∉ *S*, there is a vertex *u* ∈ *S* with *v* adjacent to *u*. The minimum cardinality of any dominating set is called the domination number of *G* and is denoted by *γ*(*G*). A dominating set *D* of a graph *G* = (*V*, *E*) is a non-split dominating set if the induced graph ⟨*V* — *D*⟩ is connected. The non-split domination number *γ*_{ns}(*G*) is the minimum cardinality of a non-split domination set. The purpose of this paper is to initiate the investigation of those graphs which are critical in the following sense: A graph *G* is called vertex domination critical if *γ*(*G* − *v*) < *γ*(*G*) for every vertex *v* in* G*. A graph *G* is called vertex non-split critical if *γ*_{ns}(*G* − *v*) < *γ*_{ns}(*G*) for every vertex *v* in *G*. Thus, *G* is *k*-*γ*_{ns}-critical if *γ*_{ns}(*G*) = *k*, for each vertex *v* ∈ *V*(*G*), *γ*_{ns}(*G *− *v*) < *k*. A graph *G* is called edge domination critical if (*G *+ *e*) < (*G*) for every edge *e* in *G*. A graph *G* is called edge non-split critical if *γ*_{ns}(*G *+ *e*) < *γ*_{ns}(*G*) for every edge *e* ∈ *G*. Thus, *G* is *k*-*γ*_{ns}-critical if *γ*_{ns}(*G*) = *k*, for each edge *e* ∈ *G*, *γ*_{ns}(*G* + *e*) < *k*. First we have constructed a bound for a non-split domination number of a subdivision graph *S*(*G*) of some particular classes of graph in terms of vertices and edges of a graph *G*. Then we discuss whether these particular classes of subdivision graph *S*(*G*) are *γ*_{ns}-critical or not with respect to vertex removal and edge addition.

**Volume 23** ▶ Number 1 ▷ Number 2 ▷ Number 3