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Applied mathematics (18)

Discrete mathematics (18)Proceedings (1)Pure mathematics (1)View MoreJournalBulletin of the Institute of Combinatorics and its Applications (ICA) (7)Fibonacci Quarterly (6)Utilitas Mathematica (3)Carpathian Journal of Mathematics (1)Electronic Notes in Discrete Mathematics (1)AuthorsBagdasar, Ovidiu (2)01f9023f-d29d-4a5e-9798-89f4c5851ab9 (1)81a26182-8169-4390-b85f-b5fad16813cb (1)a90cc4fe-8caf-4587-ab1d-ca6360cecf5b (1)afaa2e14-da18-425a-9673-04ed98ca29b9 (1)View MoreYear (Issue Date)2014-05 (4)2016-08 (2)2014 (1)2014-02 (1)2015 (1)View MoreTypesArticle (18)
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A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs

Larcombe, Peter J. (The Fibonacci Association, 2015-11)

An invariance matrix property, first observed empirically and seemingly absent from mainstream literature, is stated and established formally here. Four short, and different, proofs are given accordingly.

Closed form evaluations of some series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations

Larcombe, Peter J. (The Fibonacci Association, 2015-08)

Closed form evaluations of some infinite series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations are presented using three contrasting approaches. Known power series expansions of the trigonometric functions sin(4α) and sin(6α) each readily give a set of (four) results which are reformulated via a hypergeometric route and, additionally, using only the generating function for the Catalan sequence; the latter two methods are shown to be connected.

On the jacobsthal, horadam and geometric mean sequences

Larcombe, Peter J.; Rabago, Julius, F. T. (The Institute of Combinatorics and its Applications (ICA), 2016)

This paper, in considering aspects of the geometric mean sequence, offers new results connecting Jacobsthal and Horadam numbers which are established and then proved independently.

On a scaled balanced-power product recurrence

Larcombe, Peter J.; Fennessey, Eric J. (The Fibonacci Association, 2016-08)

A power product recurrence (due to M. W. Bunder) is extended here by the introduction of a scaling factor, and delivers a sequence whose general term closed form is derived for both degenerate and non-degenerate characteristic root cases. It is shown how recurrence parameter conditions dictate the nature of each solution type, and a fundamental link between them is highlighted together with some other observations and results.

On horadam sequence periodicity: A new approach

Larcombe, Peter J.; Fennessey, Eric J. (The Institute of Combinatorics and its Applications (ICA), 2015-01)

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.

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

Larcombe, Peter J. (The Institute of Combinatorics and its Applications (ICA), 2016-01)

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.

A non-linear identity for a particular class of polynomial families

Larcombe, Peter J.; Fennessey, Eric J. (The Fibonacci Association, 2014-02)

We state and prove a new non-linear identity for a class of polynomial families associated with integer sequences whose ordinary generating functions have quadratic governing equations with functional (polynomial) coefficients.

On the phenomenon of masked periodic Horadam sequences

Larcombe, Peter J.; Fennessey, Eric J. (The Institute of Combinatorics and its Applications (ICA), 2015)

A recently discovered phenomenon, termed masked periodicity and observed in self-repeating Horadam sequences using matrix based methods in a study by the authors elsewhere, is considered further here. This article continues the approach, identifying both governing parameters and particular behaviour types which fall naturally into three categories convenient for explanation and illustration.

Generalised Catalan polynomials and their properties

Larcombe, Peter J.; Jarvis, Frazer A.; Fennessey, Eric J. (The Institute of Combinatorics and its Applications, 2014-05)

We introduce a new type of polynomial, termed a generalised Catalan polynomial. We list essential mathematical properties and give two associated combinatorial interpretations.

Closed form evaluations of some series involving Catalan numbers

Larcombe, Peter J. (The Institute of Combinatorics and its Applications (ICA), 2014-05)

Closed form evaluations of some infinite series comprising sums of exponentiated multiples of Catalan numbers are yielded by a known expansion of the trigonometric function sin(2α), and then re-formulated independently as verification.

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