**Volume 12** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**The auxiliary equation associated with the plastic number**

*Original research paper. Pages 1—12*

A. G. Shannon, A. F. Horadam and Peter G. Anderson

Full paper (112 Kb) | Abstract

This paper looks at some of the properties of the auxiliary equation associated with the plastic number which, in turn, is related to the sequences of numbers {*P*_{n}}, {*Q*_{n}} and {*R*_{n}}, respectively, defined by

*P*_{n} = *P*_{n − 2} + *P*_{n − 3}, *n* > 3, *P*_{1} = 1, *P*_{2} = 1, *P*_{3} = 1,

*Q*_{n} = *Q*_{n − 2} + *Q*_{n − 3}, *n* > 3, *Q*_{1} = 0, *Q*_{2} = 2, *Q*_{3} = 3,

*R*_{n} = *R*_{n − 2} + *R*_{n − 3}, *n* > 3, *R*_{1} = 1, *R*_{2} = 0, *R*_{3} = 1.

The dominant root of the associated auxiliary equation is found by a contraction process related to Bernoulli’s iteration and the Jacobi—Perron Algorithm. The latter is one way of generalizing the ordinary continued fraction algorithm and an alternative way is explored which also relates to the auxiliary equations of the sequences. Various methods for reduction of the order of the cubic auxiliary equation are also considered.

**Some ***q*-binomial coefficients

*Original research paper. Pages 13—20*

A. G. Shannon

Full paper (102 Kb) | Abstract

This paper considers some *q*-extensions of binomial coefficients. Some of the results are applied to some generalized Fibonacci numbers, and others are included as ideas for further investigation, particularly, particularly into *q*-Bernoulli polynomials.

**Properties of the Sándor function**

*Original research paper. Pages 21—24*

Gabriel Mincu and Laurențiu Panaitopol

Full paper (127 Kb) | Abstract

For *x* > 0 one define the function *S*(*x*) = min{*m* ∈ ℕ | *x* ≤ *m*!}. We prove that for *x* > √13! the interval (*S*(*x*), *S*(*x*^{2})) contains at least a prime number and that for real *x*, *y* > 0 the inequality *S*(*x*) + *S*(*y*) ≥ *S*(*xy*) holds true. We also study the convergence of a couple of number series involving *S*(*x*).

**A new direction of Fibonacci sequence modification**

*Original research paper. Pages 25—32*

Krassimir T. Atanassov

Full paper (122 Kb)

**Volume 12** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4