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SubjectsDiscrete mathematics (9)Applied mathematics (6)Horadam sequences (3)Recurrent sequences (3)Periodic sequences (2)View MoreJournal

Fibonacci Quarterly (15)

AuthorsLarcombe, Peter J. (15)Fennessey, Eric J. (7)Bagdasar, Ovidiu (5)Bagdasar, Ovidiu (5) O'Neill, Sam T. (2)Year (Issue Date)2016-08 (2)2013-02-01 (1)2013-05-03 (1)2013-11-01 (1)2014-02 (1)View MoreTypesArticle (15)
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On a result of Bunder involving horadam sequences: A new proof

Larcombe, Peter J.; Bagdasar, Ovidiu; Fennessey, Eric J. (Fibonacci Association, 2014-05-02)

This note offers a new proof of a 1975 result due to M. W. Bunder which has recently been proven (inductively), extended empirically and generalized in this journal. The proof methodology, while interesting, cannot be applied realistically beyond the original order two case of Bunder dealt with here.

On a result of Bunder involving Horadam sequences: A proof and generalization

Larcombe, Peter J.; Bagdasar, Ovidiu (Fibonacci Association, 2013-05-03)

This note introduces, proves, extends empirically and generalizes a short 1975 offering of M.W. Bunder who, in this journal, gave an isolated observation involving Horadam sequences on which work has been conducted for nearly half a century.

On the number of complex horadam sequences with a fixed period

Bagdasar, Ovidiu; Larcombe, Peter J. (Fibonacci Association, 2013-11-01)

The Horadam sequence is a direct generalization of the Fibonacci numbers in the complex plane, depending on a family of four complex parameters: two recurrence coefficients and two initial conditions. Here the Horadam sequences with a given period are enumerated. The result generates a new integer sequence whose representation involves some well-known functions such as Euler's totient function φ and the number of divisors function ω.

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.

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.

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 certain series expansions of the sine function: Catalan numbers and convergence

Larcombe, Peter J.; O'Neill, Sam T.; Fennessey, Eric J. (The Fibonacci Association, 2014-08)

The appearance of Catalan numbers in certain infinite series expansions of the sine function was first reported well over a decade ago. A combination of computation and analysis is employed as we return to this topic and examine the outstanding issue of convergence for this suite of results and also for the general case expansion.

A condition for anti-diagonals product invariance across powers of $2 \times 2$ matrix sets characterizing a particular class of polynomial families

Larcombe, Peter J.; Fennessey, Eric J. (The Fibonacci Association, 2015-05)

Motivated by some recent work on a particular class of polynomial families associated with certain types of integer sequences, we formulate a sufficient condition under which the anti-diagonals products across sets of characterizing 2 × 2 matrices remain invariant as matrix power increases. Two proofs are given along with some examples.

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.

A generating function approach to the automated evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations.

Larcombe, Peter J.; O'Neill, Sam T. (The Fibonacci Association, 2018-05)

Based on a previous technique deployed in some speciﬁc low order cases, we develop an automated computational procedure to evaluate instances within a class of inﬁnite series comprising exponentiated multiples of generalized linear combinations of Catalan numbers. The methodology is explained, and new results given.

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