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

**Generalized Fibonacci and Lucas sequences with Pascal-type arrays**

*Original research paper. Pages 1—9*

Charles K. Cook and A. G. Shannon

Full paper (53 Kb) | Abstract

We re-label the Fibonacci and Lucas sequences respectively by

{*F*_{0,n}} ≡ {*F*_{n}} and {*F*_{1,n}} ≡ {*L*_{n}}

and consider

*F*_{m,n} = *F*_{m−1, n−1} + *F*_{m−1, n+1}, *m*, *n* ≥ 1,

as a generalization of the well-known identity

*L*_{n} = *F*_{n−1} + *F*_{n+1},

where

*F*_{m,n} = *F*_{m,n−1} + *F*_{m, n−2}, *m* ≥ 1, *n* > 2.

**On Steiner Loops of cardinality 20**

*Original research paper. Pages 10—22*

M. H. Armanious

Full paper (184 Kb) | Abstract

It is well known that there are five classes of sloops

** **of cardinality 16

** ” SL**(16)s” according to the number of sub-

**SL**(8)s

. In this article, we will show that there are exactly 8 classes of nonsimple sloops and 6 classes of simple sloops of cardinality 20

** “SL**(20)s”. Based on the cardinality and the number of (normal) subsloops of

** SL**(20), we will construct in section 3 all possible classes of nonsimple

** SL**(20)s and in section 4 all possible classes of simple

**SL**(20)s. We exhibit the algebraic and combinatoric properties of

**SL**(20)s to distinguish each class.

So we may say that there are six classes of **SL**(20)s having one sub-**SL**(10) and n sub-**SL**(8)s for *n* = 0, 1, 2, 3, 4 or 6. All these sloops are subdirectly irreducible having exactly one proper homomorphic image isomorphic to **SL**(2). For *n* = 0, the associated **SL**(20) is a nonsimple subdirectly irreducible having one sub-**SL**(10) and no sub-**SL**(8)s. Indeed, the associated Steiner quasigroup **SQ**(19) of this case supplies us with a new example for a semi-planar **SQ**(19), where the smallest well-known example of semi-planar squags is of cardinality 21 ” cf. “.

It is well known that there is a class of planar Steiner triple systems (**STS**(19)s) due to Doyen , where the associated planar **SL**(20) has no sub-**SL**(10) and no sub-**SL**(8). In section 4 we will show that there are other 6 classes of simple **SL**(20)s having n sub-**SL**(8)s for *n* = 0, 1, 2, 3, 4, 6, but no sub-**SL**(10)s. It is well-known that a sub-**SL**(m) of an **SL**(2m) is normal. In the last theorem of this section, we give a necessary and sufficient condition for a sub-**SL**(2) to be normal of an **SL**(2m). Accordingly, we have shown that if a sloop **SL**(20) has a sub-**SL**(10) and 12 sub-**SL**(8), then this sloop is isomorphic to the direct product **SL**(10) × **SL**(2) and if a sloop **SL**(20) has 12 sub-**SL**(8)s and no sub-**SL**(10), then this sloop is a subdirectly irreducible having exactly one proper homomorphic image isomorphic to **SL**(10). In section 5, we describe how can one construct an example for each class of smiple and of nonsimple **SL**(20)s.

**Note on ***φ*, *ψ* and *σ* functions

*Original research paper. Pages 23—24*

Krassimir T. Atanassov

Full paper (95 Kb) | Abstract

In the present remark we shall formulate and discuss some extremal problems, related to arithmetic functions

*φ*,

*ψ* and

*σ* (see, e.g.,

).

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