**Volume 30** ▶ Number 1 ▷ Number 2 (Online First)

**On the pulsating Padovan sequence**

*Original research paper. Pages 1–7*

Orhan Dişkaya and Hamza Menken

Full paper (PDF, 182 Kb) | Abstract

**Discatenated and lacunary recurrences**

*Original research paper. Pages 8–19*

Hakan Akkuş, Ömür Deveci, Engin Özkan and Anthony G. Shannon

Full paper (PDF, 944 Kb) | Abstract

**Proving the existence of Euclidean knight’s tours on n × n × ⋯ × n chessboards for n < 4**

*Original research paper. Pages 20–33*

Marco Ripà

Full paper (PDF, 1613 Kb) | Abstract

Our counterintuitive outcome follows from the observation that we can alternatively define a 2D knight as a piece that moves from one square to another on the chessboard by covering a fixed Euclidean distance of so that also the statement of Theorem 3 in [Erde, J., Golénia, B., & Golénia, S. (2012), The closed knight tour problem in higher dimensions, The Electronic Journal of Combinatorics, 19(4), #P9] does not hold anymore for such a Euclidean knight, as long as a 2 × 2 × ⋯ × 2 chessboard with at least 2

^{7}cells is given. Moreover, we construct a classical closed knight’s tour on whose arrival is at a distance of 2 from , and then we show a closed Euclidean knight’s tour on .

**Some results on geometric circulant matrices involving the Leonardo numbers**

*Original research paper. Pages 34–46*

Samet Arpacı and Fatih Yılmaz

Full paper (PDF, 316 Kb) | Abstract

**Second-order linear recurrences with identically distributed residues modulo p^{e}**

*Original research paper. Pages 47–66*

Lawrence Somer and Michal Křížek

Full paper (PDF, 257 Kb) | Abstract

**Generalization of the 2-Fibonacci sequences and their Binet formula**

*Original research paper. Pages 67–80*

Timmy Ma, Richard Vernon and Gurdial Arora

Full paper (PDF, 256 Kb) | Abstract

**Distance between consecutive elements of the multiplicative group of integers modulo n**

*Original research paper. Pages 81–99*

Steven Brown

Full paper (PDF, 314 Kb) | Abstract

In this paper, we are mostly concerned with providing formulas to count the number of gaps of a given even length in which we note . This work, presented with different notations is closely related to [5]. We prove the formulas in three steps. Although only the last step relates to the problem of gaps in the Eratosthenes sieve (see Section 3.2.2) the previous formulas may be of interest to study occurrences of defined gaps sequences.

- For a positive integer , we prove a general formula based on the inclusion-exclusion principle to count the number of occurrences of configurations
^{1}in any subset of . (see Equation (7) in Theorem 2.1). - For a square-free integer , we particularize this formula when the subset of interest is . (see Equation (11) in Theorem 3.2).
- For a prime and its primorial , we particularize the formula again to study gaps in . Given a positive integer representing a distance on the circle, we give formulas to count the number of gaps of length between elements of (see Equation (15) and Section 4.1).

In addition, we provide a formula (see Equation (27) in Theorem 5.1) to count the number of occurrences of gaps of an even length that contain exactly elements of .

^{1} A defined sequence of gaps between the elements of the subset; this is referred to as a constellation in [5].

**Some new properties of hyperbolic k-Fibonacci and k-Lucas octonions**

*Original research paper. Pages 100–110*

A. D. Godase

Full paper (PDF, 218 Kb) | Abstract

*k*-Fibonacci octonions and

*k*-Lucas octonions. We prove these properties using the identities of

*k*-Fibonacci and

*k*-Lucas numbers, which we determined previously.

**On certain arithmetical products involving the divisors of an integer**

*Original research paper. Pages 111–115*

József Sándor

Full paper (PDF, 210 Kb) | Abstract

**Sequences in finite fields yielding divisors of Mersenne, Fermat and Lehmer numbers, I**

*Original research paper. Pages 116–140*

A. M. S. Ramasamy

Full paper (PDF, 347 Kb) | Abstract

**On the characterization of rectangular duals**

*Original research paper. Pages 141–149*

Vinod Kumar and Krishnendra Shekhawat

Full paper (PDF, 181 Kb) | Abstract

**Generalized perfect numerical semigroups**

*Original research paper. Pages 150–162*

Mohammad Zmmo and Nesrin Tutaş

Full paper (PDF, 461 Kb) | Abstract

**Characterization of prime and composite numbers using the notion of successive sum of integers and the consequence in primality testing**

*Original research paper. Pages 163–169*

Fateh Mustapha Dehmeche, Douadi Mihoubi and Lahcene Ladjelat

Full paper (PDF, 579 Kb) | Abstract

**Factorial polynomials and associated number families**

*Original research paper. Pages 170–178*

Alfred Schreiber

Full paper (PDF, 195 Kb) | Abstract

**Two arithmetic functions related to Euler’s and Dedekind’s functions**

*Original research paper. Pages 179–183*

Krassimir Atanassov

Full paper (PDF, 225 Kb) | Abstract

**Metallic means and Pythagorean triples**

*Original research paper. Pages 184–194*

Chetansing Rajput and Hariprasad Manjunath

Full paper (PDF, 359 Kb) | Abstract

**Distribution of constant terms of irreducible polynomials in ℤ _{p}[x] whose degree is a product of two distinct odd primes**

*Original research paper. Pages 195–210*

Sarah C. Cobb, Michelle L. Knox, Marcos Lopez, Terry McDonald, and Patrick Mitchell

Full paper (PDF, 280 Kb) | Abstract

**Editorial Correction to “A note on a generalization of Riordan’s combinatorial identity via a hypergeometric series approach” [Notes on Number Theory and Discrete Mathematics, 2023, Volume 29, Number 3, Pages 421–425]**

*Editorial correction. Pages 211–212*

Kunle Adegoke

Editorial correction (PDF, 161 Kb)