**Volume 20** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 ▷ Number 5

**On two new means of two variables**

*Original research paper. Pages 1—9*

József Sándor

Full paper (PDF, 168 Kb) | Abstract

*A*,

*G*and

*L*denote the arithmetic, geometric resp. logarithmic means of two positive number, and let

*P*denote the Seiffert mean. We study the properties of two new means

*X*resp.

*Y*, defined by

*X*=

*A*·

*e*

^{G / P−1}and

*Y*=

*G*·

*e*

^{L /A − 1}.

**Calculating terms of associated polynomials of Perrin and Cordonnier numbers**

*Original research paper. Pages 10—18*

Kenan Kaygısız and Adem Şahin

Full paper (PDF, 150 Kb) | Abstract

**Arithmetical sequences for the exponents of composite Mersenne numbers**

*Original research paper. Pages 19—26*

Simon Davis

Full paper (PDF, 169 Kb) | Abstract

*− 1 with*

^{p}*p*prime. A lower probability for the occurrence of composite Mersenne numbers in arithmetical sequences is given.

**The decimal string of the golden ratio**

*Original research paper. Pages 27—31*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 147 Kb) | Abstract

*n*-Pulsated Fibonacci sequence

*Original research paper. Pages 32—35*

Krassimir Atanassov

Full paper (PDF, 120 Kb) | Abstract

**New modular relations for the Rogers–Ramanujan type functions of order fifteen**

*Original research paper. Pages 36—48*

Chandrashekar Adiga and A. Vanitha

Full paper (PDF, 195 Kb) | Abstract

Furthermore, we give interesting partition theoretic interpretations of these relations.

**Balancing sequences of matrices with application to algebra of balancing numbers**

*Original research paper. Pages 49—58*

Prasanta Kumar Ray

Full paper (PDF, 158 Kb) | Abstract

*n*= 0, 1, 2, …, which is both geometric (

*a*

_{n + 1}=

*ka*;

_{n}*n*= 0, 1, 2, …) and balancing (

*a*

_{n + 1}= 6

*a*

_{n}−

*a*

_{n − 1},

*a*

_{0}= 0,

*a*

_{1}= 1) admits an unique solution. In fact, the sequence is 1, λ

_{1}, λ

_{1}

^{2}, …, λ

_{1}

*, … where λ*

^{n}_{1}= 3 + √8 satisfies the balancing equation λ

^{2}− 6λ + 1. In this paper, we pose an equivalent problem for a sequence of real, nonsingular matrices of order two and show that, this problem admits an infinity of solutions, that is there exist infinitely many such sequences.

**The rectangular spiral or the n _{1} × n_{2} × … × n_{k} Points Problem**

*Original research paper. Pages 59—71*

Marco Ripà

Full paper (PDF, 589 Kb) | Abstract

*n*×

*n*× … ×

*n*Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the

*k*-dimensional

*n*

_{1}×

*n*

_{2}× … ×

*n*Points Problem. In this way, we can build a range in which, with certainty, all the best possible solutions to the problem we are considering will fall. Finally, we provide a few characteristic numerical examples in order to appreciate the fineness of the result arising from the particular approach we have chosen.

_{k}**The Fibonacci sequence and the golden ratio in music**

*Original research paper. Pages 72—77*

Robert van Gend

Full paper (PDF, 414 Kb) | Abstract

**Number Theory: A Historical Approach**

*Book review. Page 78*

A. G. Shannon

Book review (PDF, 44 Kb)

**Volume 20** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 ▷ Number 5