**Volume 11** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**On the structure of certain counting polynomials**

*Original research paper. Pages 1—11*

R. Valenza

Full paper (PDF, 135 Kb) | Abstract

^{(k)}(

*s*) = Σ

*α*/

_{n}*n*(

^{s}*k*≥ 2) of the Riemann zeta function that arises from a modification of its classical Euler product expansion, for the most part here concentrating on the case

*k*= 2. The associated coefficients

*α*correspond to a counting problem that may be addressed via a family of multivariable generating functions. Examples computed via symbolic manipulation suggest a recursive structure for these functions, which we prove. With this result in hand, the calculation of the α

_{n}_{n}may be facilitated by a more efficient, double modular algorithm, as worked out in a detailed example. We conclude with some observations and questions for the case

*k*> 2.

**On two new 2-Fibonacci sequences**

*Original research paper. Pages 12—16*

Krassimir T. Atanassov

Full paper (PDF, 1604 Kb)

**Fermat and Mersenne numbers**

*Original research paper. Pages 17—24*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 114 Kb) | Abstract

*F*= 2

_{n}^{2n}+ 1) and Mersenne numbers (

*M*= 2

_{m}*− 1),*

^{m}*m*odd, are compared on the basis of integer structure, using the modular rings Z

_{4}and Z

_{6}. The two numbers fall in different classes and this results in different composite row structures and different potentials for the formation of primes. The constraints on 2

*and the right end digits for*

^{n}*F*result in fewer numbers over a given range than those for

_{n}*M*. This is shown with two functions, which link the two numbers and show that

_{m}*F*= (2

_{n}^{x2 + y2 − 1}+ 1): for primes

*y*=

*n*, but when

*n*> 4,

*y*≠

*n*.

**On the sum of equal powers of the first n terms of an arbitrary arithmetic progression. II**

*Original research paper. Pages 25—28*

Peter Vassilev and Mladen Vassilev-Missana

Full paper (PDF, 1146 Kb)