A few thoughts on the aesthetics of mathematics in research and teaching.

Mathematical aesthetic, having a variety of forms, is commonly experienced by mathematicians as a genuine reality and is available at every level of study. This short essay in hopefully moving beyond standardised hermeneutic critique attests to its authenticity through the words of some mathematical proponents, and reminds us that it should be promoted to children and students as a phenomenon that is as accessible as it is incontestable.N/

Horadam sequences: A survey update and extension.

We give an update on work relating to Horadam sequences that are generated by a general linear recurrence formula of order two. This article extends a first ever survey published in early 2013 in this Bulletin, and includes coverage of a new research area opened up in recent times.N/

On the notion of mathematical genius: rhetoric and reality.

The existential hypothesis of mathematical ‘genius’ rests irrefutably with the affirmative. We can’t all be one of course—that sublime solopreneur in creativity—but we may still contribute to research as lesser mortals. This short essay attempts to explore what the notion means to both us and non-mathematicians.N/

Alwyn Francis Horadam, 1923-2016: A personal tribute to the man and his sequence

Having received news of the passing of Alwyn Horadam this last July, I was determined that I should write something in his honour in which my own contact with him is described and combined with some introductory details on what I feel is his major endowment to the community of mathematicians - the so called and pre-eminent Horadam sequence whose specialisations thereof are great in number

A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power

We obtain an existing 2004 result of J. McLaughlin which gives explicit entries for a general dimension 2 matrix raised to an arbitrary power. Our formulation employs so called Catalan polynomials related to the crucial parameter of McLaughlin’s statement, and is a new one running along a different line of argument

On sequence-based closed form entries for an exponentiated general 2×22 \times 2 matrix: A re-formulation and an application.

Closed form entries for an exponentiated (and arbitrary) 2 ⇥ 2 matrix are established here, and expressed in terms of a specialized Horadam sequence; two proofs of the result are given accordingly, along with examples and observations derived therefrom. The result o↵ers a new formulation of a general class of polynomial families associated with sequences whose ordinary generating functions are governed by quadratic equations.N/

On horadam sequence periodicity: A new approach

A so called Horadam sequence is one delivered by a general second order recurrence formula with arbitrary initial conditions. We examine aspects of self-repeating Horadam sequences by applying matrix based methods in new ways, and derive some conditions governing their cyclic behaviour. The analysis allows for both real and complex sequence periodicity

On generalised multi-index non-linear recursion identities for terms of the Horadam sequence.

We state and prove a non-linear recurrence identity for terms of the so called Horadamsequence,andthenofferitsgeneralisationwhichisavailablefromthesamemethodology. We illustrate how the overarching idea may be used to sequentially produce extended versions, each in turn with an extra level of non-linearity and term index complexity. These identities can all be captured in matrix determinant form.N/

A new non-linear recurrence identity class for Horadam sequence terms.

We state, and prove by a succinct matrix method, a non-linear recurrence identity class for terms of the so called Horadam sequence. A particular instance was established (in equivalent form) over half a century ago by A.F. Horadam, which provides a starting point for the discussion and an introduction to our formulation technique.N/