**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

Let *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

In this paper, we calculate terms of associated polynomials of Perrin and Cordonnier numbers by using determinants and permanents of various Hessenberg matrices. Since these polynomials are general forms of Perrin and Cordonnier numbers, our results are valid for the Perrin and Cordonnier numbers.

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

*Original research paper. Pages 19—26*

Simon Davis

Full paper (PDF, 169 Kb) | Abstract

Arithmetical sequences for the exponents of composite Mersenne numbers are obtained from partitions into consecutive integers, and congruence relations for products of two Mersenne numbers suggest the existence of infinitely many composite integers of the form 2^{p} − 1 with *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

The decimal expansion of the Golden Ratio is examined through the use of various properties of the Fibonacci numbers and some exponential functions.

*n*-Pulsated Fibonacci sequence

*Original research paper. Pages 32—35*

Krassimir Atanassov

Full paper (PDF, 120 Kb) | Abstract

A new type of Fibonacci sequence is introduced and explicit formulas for the form of its members are formulated and proved. It is an extension of the special Fibonacci sequence, introduced in

and called a Pulsated Fibonacci sequence.

**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

In this paper, we establish two modular relations for the Rogers–Ramanujan–Slater functions of order fifteen. These relations are analogues to Ramanujan’s famous forty identities for the Rogers–Ramanujan functions.

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

It is well known that, the problem of finding a sequence of real numbers an, *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}^{n}, … where λ_{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

A generalization of Ripà’s square spiral solution for the *n* × *n* × … × *n* Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the *k*-dimensional *n*_{1} × *n*_{2} × … × *n*_{k} 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.

**The Fibonacci sequence and the golden ratio in music**

*Original research paper. Pages 72—77*

Robert van Gend

Full paper (PDF, 414 Kb) | Abstract

This paper presents an original composition based on Fibonacci numbers, to explore the inherent aesthetic appeal of the Fibonacci sequence. It also notes the use of the golden ratio in certain musical works by Debussy and in the proportions of violins created by Stradivarius.

**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