There have been a number of articles on the relation between the terms of the Fibonacci and Lucas sequences and how they are closely related to trigonometric and hyperbolic functions and their properties [1]. This article is based on other integer sequences. It sets out to determine other pairs of such sequences that have the same relation as the Fibonacci and Lucas have to each other. So we shall be concerned with second order recurrence relations with constant coefficients:

and pairs of sequences (un) and (vn) that each satisfy it. We seek a condition that ensures the pair of sequences behave as the Fibonacci-Lucas pair behave.

Albrecht Dürer's mysterious engraving Melencolia I (Figure 1) has always intrigued both art critics and mathematicians. Among art critics, according to Campbell Dodgson [1], “The literature on the Melancholia is more extensive than that on any other engraving by Dürer” (he adds “the statement would probably remain true if the last two words were omitted”). Mathematicians, if disconcerted by the association between mathematics and melancholy, have been fascinated by the objects appearing in the print, such as the polyhedron occurring on the left of the engraving and—the subject of this note—the magic square in the upper right-hand corner (Figure 2).

Older readers will remember using a slide rule to carry out multiplication and division in the days before calculators. There was also a graphical tool called a nomogram, which was used for evaluating certain functions. Although such devices have little practical use nowadays, I was interested in investigating which classes of functions with two independent variables could be calculated using this type of device.

Figure 1 shows how a slide rule was used to multiply 3 by 4 (or equivalently to divide 12 by 4). To do this, you align the 1 on the upper scale with the 3 on the lower scale and then read off the value below the 4 on the upper scale to get 12 (or, for division, you line up the 4 above the 12 and read off the answer 3 on the lower scale).

Like Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers are a fertile ground for creativity and exploration. They also have interesting applications to combinatorics [1], especially to the study of lattice paths [2, 3], as we will see shortly.

Pell numbers Pn and Pell-Lucas numbers Qn are often defined recursively [4, 5]:

where n ≥ 3. They can also be defined by Binet-like formulas:

The motivation to write this paper arose out of the following problem which was posed in a recent mathematical olympiad:

Given a polynomial P(X) with integer coefficients, show that there exist non-zero polynomials Q(X), R(X) with integer coefficients such that P(X)Q(X) is a polynomial in X2 and P(X)R(X) is a polynomial in X3.

For instance, if P(X) = 2 − 5X + 3X2 + 12X3, then we notice that Q (X) = 2 + 5X + 3X2 − 12X3 serves the purpose for the first part, viz.

A moment's thought makes it fairly evident that this trick easily solves the first part of the problem for a general polynomial P(X).

Archimedes developed a method for calculating the area of a segment of a parabola. This result was outstanding in mathematical antiquity. The idea and the method were not developed further until nearly 1800 years later. In our article his method is presented in modem notation using modem concepts. We show that the method generalises to curves of higher order. Many of Archimedes' arguments can be used without any adjustments and thus show the strength of his original approach.

The mathematician from Syracuse, Archimedes (born about 287 BC, died 212 BC), developed a method for the calculation of the area of a segment of the parabola.

Many readers will be familiar with the sequence of Catalan numbers {Cn: n ≥ 0} and the formula

with its alternative form

These can be proved by using recurrence relations, generating functions or André's reflection principle. A good reference for all of these methods is Martin Griffiths' book [1].

However, none of these approaches strikes me as being naturally combinatorial. A formula such as (1) is often derived by making a list of all the ways of doing something, and then subdividing this list into classes of equal size, so that either one class consists entirely of ‘valid’ cases or there is exactly one ‘valid’ case in each list.

Every student of calculus learns that one typically solves a differential equation by integrating it. However, as Euler showed in his 1758 paper (E236), Exposition de quelques paradoxes dans le calcul intégral (Explanation of certain paradoxes in integral calculus) [1], there are differential equations that can be solved by actually differentiating them again. This initially seems paradoxical or, as Euler describes it in the introduction of his paper:

Here I intend to explain a paradox in integral calculus that will seem rather strange: this is that we sometimes encounter differential equations in which it would seem very difficult to find the integrals by the rules of integral calculus yet are still easily found. not by the method of integration. but rather in differentiating the proposed equation again; so in these cases, a repeated differentiation leads us to the sought integral.

A pleasurable aspect of mathematics and its teaching is to review the diversity of ways in which theorems are proved. Especially in elementary branches, there are various kinds of proof: using (or avoiding) spatial geometry, analytic or coordinate geometry, common algebra, vectors, abstract algebras, matrices, determinants, the differential and integral calculus, and maybe mixtures thereof. Further, sometimes a proof of one kind is elegant while another is clumsy, or one proof of a theorem suggests why it follows while another proof is not perspicuous. There is also the question of whether a proof is direct or indirect (for example, proofby contradiction).

This article is concerned with two sequences and the matrices they induce. The first, the Derangement sequence is well known to readers of the Gazette. It has been written about and its properties derived in many ways and on many previous occasions [1 – 11]. In contrast the second sequence, the Deadend sequence, is not well known – and certainly not the name since I made that up while writing this article. However, I want to explore these sequences in a single, systematic way. For the former it gives a fresh account of its properties, and for the latter it sets out corresponding results. The surprise is that they tum out to be closely related; surprising since their very different definitions mask such a connection. More than this, most of these properties may be established in the most fundamental way, based on elegant enumerative arguments. So they are free of specialised expertise or technique and may be enjoyed just as they are.

This paper is about the classification of frieze groups. A frieze is a decorative strip of paper (or wood, stone,…) on which a pattern is produced by the periodic repetition of a picture along the strip. A symmetry of a frieze is an isometry of the plane that leaves the pattern unchanged, and a frieze group is the group of symmetries of some frieze. A popular exercise is to start with a given frieze and then try to identify its frieze group. However, in order to do this we need to know that there are only seven possible frieze groups, and what these groups are. Seven friezes, with different frieze groups, are illustrated below, in such a way that each pattern is invariant under the same translation, namely x → x + 1. Our task is to show that (up to a change in the motif, and a simple change of coordinates in the plane) these are the only frieze patterns.

As I write (Spring 2012) I am aware that I have been a member of the Mathematical Association for fifty years. I joined during the annual conference in 1962 when Dr Combridge was the main figure in the society. The highlight was a session at the conference (at King's College, London) by Geoffrey Matthews showing how matrices might be introduced into the school curriculum. Some treated this as hilarious, others as a signpost to the future. My friend Frank Budden and I regarded it as a premature ploy, which, we thought, might not tum out to be such a good idea. We subsequently wrote a successful book Mathematics through Geometry (1964) arguing in detail why the modem mathematics revolution—which was then gathering pace—might end in tears. We identified spatial imagination as the heartland of mathematics, and contrasted it with a denial of spatial imagination in the abstract calculi which were all the vogue at the time.