**Volume 29** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Tony Shannon on ↑(39)**

*Editorial. Page i *

**On a generalization of dual-generalized complex Fibonacci quaternions**

*Original research paper. Pages 635–646*

Elif Tan and Umut Öcal

Full paper (PDF, 199 Kb) | Abstract

**New approaches of ( q,k)-Fibonacci–Pell sequences via linear difference equations. Applications**

*Original research paper. Pages 647–669*

Irene Magalhães Craveiro, Elen Viviani Pereira Spreafico and Mustapha Rachidi

Full paper (PDF, 317 Kb) | Abstract

*q*,

*k*)-Fibonacci–Pell sequences via linear difference equations of order 2 with variable coefficients, and explore some of their new properties. More precisely, our results are based on two approaches, namely, the determinantal and the nested sums approaches, and their closed relations. As applications, we investigate the

*q*-analogue Cassini identities and examine a pair of Rogers–Ramanujan type identities.

**Identities for Fibonacci and Lucas numbers**

*Original research paper. Pages 670–681*

George Grossman, Aklilu Zeleke and Xinyun Zhu

Full paper (PDF, 169 Kb) | Abstract

**Almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers**

*Original research paper. Pages 682–694*

Ahmet Tekcan and Esra Zeynep Türkmen

Full paper (PDF, 230 Kb) | Abstract

**On sums with generalized harmonic numbers via Euler’s transform**

*Original research paper. Pages 695–704*

Neşe Ömür, Sibel Koparal and Laid Elkhiri

Full paper (PDF, 218 Kb) | Abstract

**Perfect numbers, Wieferich primes and the solutions of **

*Original research paper. Pages 705–712*

Gabriel Guedes and Ricardo Machado

Full paper (PDF, 302 Kb) | Abstract

**Lower bounds on expressions dependent on functions φ(n), ψ(n) and σ(n)**

*Original research paper. Pages 713–716*

Stoyan Dimitrov

Full paper (PDF, 247 Kb) | Abstract

connecting , and -functions are formulated and proved.

**Note on the general monic quartic equation**

*Original research paper. Pages 717–723*

Ioana Petkova

Full paper (PDF, 160 Kb) | Abstract

**Binary expansions of prime reciprocals**

*Original research paper. Pages 724–736*

Brenda Navarro-Flores, José M. González-Barrios and Raúl Rueda

Full paper (PDF, 257 Kb) | Abstract

**The 2-adic valuation of the general degree-2 polynomial in 2 variables**

*Original research paper. Pages 737–751*

Shubham

Full paper (PDF, 264 Kb) | Abstract

**Notes on generalized and extended Leonardo numbers**

*Original research paper. Pages 752–773*

Anthony G. Shannon, Peter J.-S. Shiue and Shen C. Huang

Full paper (PDF, 359 Kb) | Abstract

*n*-Rooks and *n*-queens problem on planar and modular chessboards with hexagonal cells

*Original research paper. Pages 774–788*

Eduard C. Taganap and Rainier D. Almuete

Full paper (PDF, 2044 Kb) | Abstract

**New Fibonacci-type pulsated sequences**

*Original research paper. Pages 789–793*

Lilija Atanassova and Velin Andonov

Full paper (PDF, 165 Kb) | Abstract

numbers. For example, for such that

and

where is an -derangement number.

**The Wiener, hyper-Wiener, Harary and SK indices of the P(Z_{pk·qr}) power graph**

*Original research paper. Pages 794–803*

Volkan Aşkin

Full paper (PDF, 758 Kb) | Abstract

In this article we focus more on the indices of power graph by Wiener, hyper-Wiener, Harary and *SK* the definition of the power graph is presented and the results and theorems which we need in our discussion are provided in the introduction. Finally, the main point of the article is that we calculate the Wiener, hyper-Wiener, Harary and *SK* indices of the power graph corresponding to the vertex . These are as follows: are distinct primes and are nonnegative integers.

**On generalized hyperharmonic numbers of order **

*Original research paper. Pages 804–812*

Sibel Koparal, Neşe Ömür and Laid Elkhiri

Full paper (PDF, 282 Kb) | Abstract

numbers. For example, for such that

and

where is an -derangement number.

**On a modification of Set(n)**

*Original research paper. Pages 813–819*

Krassimir T. Atanassov and József Sándor

Full paper (PDF, 300 Kb) | Abstract

**Nontrivial lower bounds for the p-adic valuations of some type of rational numbers and an application for establishing the integrality of some rational sequences**

*Original research paper. Pages 820–826*

Bakir Farhi

Full paper (PDF, 228 Kb) | Abstract

*p*-adic valuation (where

*p*is a given prime number) of some type of rational numbers involving harmonic numbers. Then we use our estimate to derive the integrality of some sequences of rational numbers, which cannot be seen directly from their definitions.

**The t-Fibonacci sequences in the 2-generator p-groups of nilpotency class 2**

*Original research paper. Pages 827–841*

Elahe Mehraban, Ömur Deveci and Evren Hincal

Full paper (PDF, 320 Kb) | Abstract

*p*-groups of nilpotency class 2. We will discuss the lengths of the periods of the

*t*-Fibonacci sequences in these groups.

**Some geometric properties of the Padovan vectors in Euclidean 3-space**

*Original research paper. Pages 842–860*

Serdar Korkmaz and Hatice Kuşak Samancı

Full paper (PDF, 743 Kb) | Abstract

*Geogebra*program. In addition, the first ten Padovan numbers and Padovan vectors were calculated using the Binet formulas and shown as points and vectors in three-dimensional space. According to the Padovan vectors found, the Padovan curve was drawn in space for the first time by using the curve fitting feature of the

*Geogebra*program. Thus, with our study, a geometric approach to Padovan number sequences was brought for the first time.

**On tertions and other algebraic objects**

*Original research paper. Pages 861–880*

Krassimir T. Atanassov

Full paper (PDF, 267 Kb) | Abstract

*This volume of the International Journal “Notes on Number Theory and Discrete Mathematics” is published with the financial support of the Bulgarian National Science Fund, Grant Ref. No. KP-06-NP4/30/2022.*