#### Publications

Isnaini, U, Melham, R & Toh, PC 2019, 'The number of representations of a positive integer by triangular, square and decagonal numbers', *Bulletin of the Korean Mathematical Society*, vol. 56, no. 5, pp. 1143-1157.View/Download from: UTS OPUS or Publisher's site

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© 2019 Korean Mathematical Society. Let TaDb(n) and TaDb(n) denote respectively the number of representations of a positive integer n by a(x2 - x)/2 + b(4y2 - 3y) and a(x2 - x)/2 + b(4y2 - y). Similarly, let SaDb(n) and SaD’b(n) denote respectively the number of representations of n by ax2 + b(4y2 - 3y) and ax2 + b(4y2 - y). In this paper, we prove 162 formulas for these functions.

Melham, RS 2018, '12 Two-parameter families of reciprocal sums of products of the sine and cosine functions', *Fibonacci Quarterly*, vol. 56, no. 4, pp. 329-333.View/Download from: UTS OPUS

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© 2018 The Fibonacci Association. All rights reserved. In this paper, we give closed forms for 12 two-parameter families of finite sums. In each of the aforementioned 12 families of finite sums, the denominator of the summand consists of a product of the sine or cosine functions, and the length of this product can be made as large as we please.

Melham, RS 2018, 'Closed forms for 10 families of finite sums of fractional generalized Fibonacci products', *Fibonacci Quarterly*, vol. 56, no. 4, pp. 290-295.View/Download from: UTS OPUS

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© 2018 The Fibonacci Association. All rights reserved. In this paper, we present closed forms for 10 families of finite sums where the denominator of the summand is a product of generalized Fibonacci numbers. In each of these 10 families, the product in the denominator of the summand is arbitrarily long.

Melham, RS 2018, 'Closed formulas for finite sums of fractional expressions that involve the sine and cosine functions', *Fibonacci Quarterly*, vol. 56, no. 4, pp. 360-362.View/Download from: UTS OPUS

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© 2018 The Fibonacci Association. All rights reserved. In this paper, we give closed formulas for a number of fractional expressions that involve the sine and cosine functions. In each case, the argument of sine/cosine is in arithmetic progression or geometric progression.

Melham, RS 2018, 'Closed formulas for finite sums of weighted fractional generalized fibonacci products', *Fibonacci Quarterly*, vol. 56, no. 2, pp. 167-176.View/Download from: UTS OPUS

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© 2018 Fibonacci Association. All rights reserved. In this paper, we present closed formulas for finite sums of fractions involving weighted products of generalized Fibonacci numbers. There are two significant features that characterize the main results. First, the product in the denominator of each summand can be arbitrarily long. Second, in each summand that we consider, there is a so-called weight term. The weight term occurs either in the numerator or the denominator of the summand.

Melham, RS 2018, 'Further closed forms for finite sums of weighted products of generalized fibonacci numbers', *Fibonacci Quarterly*, vol. 56, no. 1, pp. 3-9.View/Download from: UTS OPUS

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© 2018 Fibonacci Association. All rights reserved. The finite sum n 2−iFi−1 = 1 −Fn+2 2n, i=1 occurs in Section 9.1 of Knott [1], and is the inspiration for the present paper. We refer to the term 2−iin the summand as the weight term. Here, we present seven families of such finite sums that we believe to be new. In each of these seven families, the product that defines the summand can be made arbitrarily long. The sequences that we employ are generalizations of the Fibonacci/Lucas sequences.

Melham, RS 2018, 'Further closed forms for finite sums of weighted products of the sine and cosine functions', *Fibonacci Quarterly*, vol. 56, no. 1, pp. 38-42.View/Download from: UTS OPUS

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© 2018 Fibonacci Association. All rights reserved. In this paper, we present closed forms for six families of finite sums of weighted products of the sine/cosine functions. In each finite sum that we define, the summand contains a product of trigonometric functions, and the length of this product can be made as large as we please. A special case of one of our main results is the sumi=1n2 cos12icos i cos(i − 3) = cos 1 −cosn cos(n + 1) . 2ncosn2 Here the weight term in the summand is2 cos12i.

Melham, RS 2018, 'Sums of reciprocals of weighted products of the sine and cosine functions', *Fibonacci Quarterly*, vol. 56, no. 2, pp. 99-105.View/Download from: UTS OPUS

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© 2018 Fibonacci Association. All rights reserved. In this paper, we define 12 families of finite sums that involve the sine/cosine functions. Four of these families are parametrized by j, and the remaining eight families are parametrized by j and k. In each of the aforementioned 12 families, the denominator of the summand contains a product of sine or cosine functions, and the length of this product is governed by the parameter j. As such, the length of the product in question can be made as large as we please. In each of the 12 families of finite sums that we consider, there is a so-called weight term in the summand. For instance, in S4 (defined in Section 2), the weight term is (2 cos1j)i

Melham, RS 2017, 'Closed forms for certain fibonacci type sums that involve second order products', *Fibonacci Quarterly*, vol. 55, no. 3, pp. 195-200.View/Download from: UTS OPUS

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In this paper, we present closed forms for certain finite sums in which the summand is a product of generalized Fibonacci numbers. We present our results in the form of six theorems that feature a generalized Fibonacci sequence {Wn}, and an accompanying sequence {Wn}- We add a further layer of generalization to our results with the use of three parameters s, k, and m. The inspiration for this paper comes from a website of Knott that lists so-called order 2 summations involving the Fibonacci and Lucas numbers. Probably the most well-known of these summations is σni=1Fi2=FnFn+1.

Melham, RS 2017, 'Closed forms for finite sums of weighted products of generalized Fibonacci numbers', *Fibonacci Quarterly*, vol. 55, no. 2, pp. 99-104.View/Download from: UTS OPUS

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In this paper, we present closed forms for certain finite sums of weighted products of generalized Fibonacci numbers. Indeed, we present seven multi-parameter families of such finite sums, all of which we believe to be new. In each of these families, the number of factors in the summand is governed by the size of the integer parameter j ≥ 1, and can be made as large as we please. We present our main results in terms of sequences that generalize the Fibonacci/Lucas numbers. Consequently, each of our main results can be specialized to involve the Fibonacci/Lucas numbers. For instance, as a consequence of one of our main results, it follows that (Equation presented) Here the weight term in the summand is 2i-1.

Melham, RS 2017, 'Closed forms for finite sums of weighted products of the sine and cosine functions', *Fibonacci Quarterly*, vol. 55, no. 2, pp. 123-128.View/Download from: UTS OPUS

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In this paper, we present closed forms for eight finite sums of weighted products of the sine/cosine functions. In each finite sum that we define, the number of factors in the summand is governed by the size of the integer parameter j ≥ 1, and can be made as large as we please. As a consequence of one of our main results, it follows that (Equation presented) Here the weight term in the summand is (2 cos 1)1-1.

Melham, RS 2017, 'Sums of certain products of fibonacci and Lucas numbers-part III', *Fibonacci Quarterly*, vol. 55, no. 3, pp. 229-234.View/Download from: UTS OPUS

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For the Fibonacci numbers, the summation formula σ n k=1 F k 2 =F n F n+1 is well-known. Its charm lies in the fact that the right side is a product of terms from the Fibonacci sequence. In the earlier paper [5], the author presents similar formulas where, in each case, the right side consists of arbitrarily long products of an even number of distinct terms from the Fibonacci sequence. The formulas in question contain several parameters, and this contributes to their generality. In this paper, we provide additional results of a similar nature where the right side consists of arbitrarily long products of an odd number of distinct terms from the Fibonacci sequence. Most of the results that we present apply to a sequence that generalizes both the Fibonacci and Lucas numbers.

Melham, RS 2016, 'A two parameter pell diophantine equation that generalizes a fibonacci classic', *Fibonacci Quarterly*, vol. 54, no. 2, pp. 112-117.

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In this paper, we present the results of our investigation into a two parameter Pell Diophantine equation. With certain constraints on the two parameters, we present the positive integer solutions of the Pell equation in question. Indeed, assuming these constraints, we express the positive integer solutions in terms of a second order recurring sequence. For certain values of the parameters, the Pell equation in question reduces to a classic Pell equation, whose solutions are expressed in terms of Fibonacci and Lucas numbers.

Melham, RS 2016, 'Closed forms for finite sums in which the denominator of the summand is a product of trigonometric functions', *Fibonacci Quarterly*, vol. 54, no. 3, pp. 196-203.

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In this paper, we present closed forms for certain finite sums. In each case, the denominator of the summand is a product of sine or cosine functions. Furthermore, in each case, the arguments of the trigonometric functions in the denominator of the summand increase in arithmetic progression.

Melham, RS 2016, 'More new algebraic identities and the Fibonacci summations derived from them', *Fibonacci Quarterly*, vol. 54, no. 1, pp. 31-43.

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In this paper, we introduce certain algebraic identities that we believe are new. For each of these algebraic identities, one side telescopes when summed. A link to the Fibonacci/Lucas numbers then facilitates the derivation of closed forms for reciprocal series that involve the Fibonacci/Lucas numbers. The term that defines the denominator of each summand contains squares of Fibonacci related numbers, with subscripts in arithmetic progression.

Melham, RS 2016, 'New identities satisfied by powers of Fibonacci and Lucas numbers', *Fibonacci Quarterly*, vol. 54, no. 4, pp. 296-303.

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The impetus for this research came from previous work of the author and others. This work centered around finding generalizations of the identities Fn+12+ Fn2= F2n+1, Fn+13+ Fn3- Fn-13= F3n, and of their higher power analogues. The main result in this paper represents an addition to the literature of such identities. Specifically, the main result is an identity satisfied by mth powers of Fibonacci numbers in which the subscripts of the Fibonacci numbers involved are arbitrarily spaced. From this main result, additional (similar) identities that involve the Fibonacci/Lucas numbers arise as so-called dual identities.

Melham, RS 2016, 'On a classical Fibonacci identity of Aurifeuille', *Fibonacci Quarterly*, vol. 54, no. 1, pp. 19-22.

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In this paper, we present identities that are analogous to a classical Fibonacci identity of Aurifeuille. Aurifeuille's identity gives certain factors of L5n, n odd. The analogues of Aurifeuille's identity that we present involve pairs of sequences that generalize the Fibonacci and Lucas sequences.

Melham, RS 2016, 'On a generalized Pell equation studied by Euler and Sadek', *Fibonacci Quarterly*, vol. 54, no. 1, pp. 49-54.

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In this paper, we follow the path set forth by Euler and Sadek in their study of a generalized Pell equation. Euler and Sadek effectively studied the positive rational solutions of the Pell equation in question. Here, we study the positive integer solutions of this equation. Indeed, by imposing additional conditions on certain parameters, we demonstrate that the positive rational solutions produced by a certain system of Euler and Sadek turn out to be positive integer solutions. Furthermore, the positive integer solutions produced by this system are the only positive integer solutions of the Pell equation in question.

Melham, RS 2016, 'On the positive integer points of certain two parameter families of hyperbolas', *Fibonacci Quarterly*, vol. 54, no. 3, pp. 247-252.

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In this paper, in Sections 2-4, we consider the problem of finding all the positive integer points of a certain two parameter family of quadratic Diophantine equations. Geometrically, this family of quadratic Diophantine equations is a two parameter family of hyperbolas. We proceed by linking the positive integer points of this family of hyperbolas to the positive integer points of a two parameter family of Pell equations. Along the way, we prove a theorem that gives the fundamental solution of the two parameter family of Pell equations in question. In Section 5, we summarize our findings regarding the positive integer points of another two parameter family of quadratic Diophantine equations.

Melham, RS 2016, 'Two algebraic identities and the alternating Fibonacci sums produced by them', *Fibonacci Quarterly*, vol. 54, no. 2, pp. 154-159.

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In this paper, we present two algebraic identities that we believe are new. These algebraic identities produce, via the telescoping effect, closed forms for a host of finite sums that involve the Fibonacci/Lucas numbers. All the finite sums contained herein are alternating in nature.

Melham, R 2015, 'Reciprocal series of squares of Fibonacci related sequences with subscripts in arithmetic progression', *Journal of Integer Sequences*, vol. 18, no. 8, pp. 1-13.

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In this paper, we derive closed forms for reciprocal series, both finite and infinite, that involve Fibonacci numbers. The term that defines the denominator of each summand generates squares of Fibonacci related numbers with subscripts in arithmetic progression. Our method employs certain algebraic identities that we believe are new. These identities exhibit the telescoping effect when summed.

Melham, RS 2015, 'On certain families of finite reciprocal sums that involve generalized Fibonacci numbers', *Fibonacci Quarterly*, vol. 53, no. 4, pp. 323-334.

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In this paper, we find closed forms, in terms of rational numbers, for certain finite sums. Our most general results are for finite sums where the denominator of the summand is a product of terms from a sequence that generalizes both the Fibonacci and Lucas numbers.

Melham, RS 2014, 'Finite reciprocal sums involving summands that are balanced products of generalized fibonacci numbers', *Journal of Integer Sequences*, vol. 17, no. 6, pp. 1-11.View/Download from: UTS OPUS

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© 2014, University of Waterloo. All rights reserved. In this paper we find closed forms, in terms of rational numbers, for certain finite sums. The denominator of each summand is a finite product of terms drawn from two sequences that are generalizations of the Fibonacci and Lucas numbers.

Melham, R 2013, 'Finite Sums that involve Reciprocals of Products of Generalized Fibonacci Numbers', *Integers*, vol. 13.View/Download from: UTS OPUS

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In this paper we find closed forms for certain finite sums. In each case the denominator of the summand consists of products of generalized Fibonacci numbers. Furthermore, we express each closed form in terms of rational numbers.

Melham, R 2012, 'On Finite Sums of Good and Shar that involve Reciprocals of Fibonacci Numbers', *Integers*, vol. 12.View/Download from: UTS OPUS

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Recently Nathaniel Shar presented a finite sum, involving the Fibonacci numbers, that generalizes a classical result first considered by I. J. Good and others. In this paper we provide a generalization of Shars sum. Furthermore, we give an analogue for the Lucas numbers. Finally we note that our generalization of Shars sum and its analogue for the Lucas numbers carry over to certain one parameter generalizations of the Fibonacci and Lucas numbers.

Melham, R 2011, 'ON PRODUCT DIFFERENCE FIBONACCI IDENTITIES', *Integers*, vol. 11, pp. 1-8.View/Download from: UTS OPUS or Publisher's site

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Simson's identity is a well-known Fibonacci identity in which the difference of certain order 2 products has a particularly pleasing form. Other old and beautiful identities of a similar nature are attributed to Catalan, Gelin and Cesaro, and Tagiuri. Catalan's identity can be described as a family of product difference Fibonacci identities of order 2 with 1 parameter. In Section 2 of this paper we present four families of product difference Fibonacci identities that involve higher order products. Being self-dual, each of these families may be regarded as a higher order analogue of Catalan's identity. We also state two conjectures that give the form of similar families of arbitrary order. In the final section we give other interesting product difference Fibonacci identities.

Berrizbeitia, P, Luca, F & Melham, R 2010, 'On a compositeness test for (2p+1)/3', *Journal of Integer Sequences*, vol. 13, no. 1, pp. 1-6.View/Download from: UTS OPUS

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n this note, we give a necessary condition for the primality of (2p+1)/3.

Melham, R 2010, 'Analogues of Jacobi's two-square theorem: an informal account', *Integers*, vol. 10, pp. 83-100.View/Download from: UTS OPUS

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Jacobi's two-square theorem states that the number of representations of a positive integer k as a sum of two squares, counting order and sign, is 4 times the surplus of positive divisors of k congruent to 1 modulo 4 over those congruent to 3 modulo 4. In this paper we give numerous identities, each of which yields an analogue of Jacobi's result. Our identities are drawn from a much larger list, and involve polygonal numbers. The formula for the nth k-gonal number is

Melham, R 2010, 'More on combinations of higher powers of fibonacci numbers', *The Fibonacci Quarterly*, vol. 48, no. 4, pp. 307-311.View/Download from: UTS OPUS

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The Fibonacci identity belongs to a family of identities where each identity contains only one product on the right side. In this paper we give this family together with two other such families. We also state two conjectures that give the form of similar identities. Finally, we give the expansions of L;m and F;;m in terms of Lucas numbers with even subscripts.

Melham, R 2010, 'On certain combinations of higher powers of Fibonacci numbers', *The Fibonacci Quarterly*, vol. 48, no. 3, pp. 256-259.View/Download from: UTS OPUS

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We present identities that we feel can be regarded as higher order analogues of the well-known identity F; + F;+l = Hn+l. We give three theorems corresponding to the powers 4, 6, and 8. We also state two conjectures that give the form of similar identities that involve higher powers.

Melham, R 2010, 'On the representation of certain reals via the golden ratio', *The Fibonacci Quarterly*, vol. 48, no. 2, pp. 150-160.View/Download from: UTS OPUS

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Taking the reciprocal of the golden ratio and summing its non-negative integer powers, we obtain a series that converges. We then consider series obtained by striking out terms of this series, proving key theorems about them and the real numbers to which hey converge. Finally, we preassign two-parameter families of real numbers related to the Fibonacci numbers and give their series expansions.

Melham, R 2009, 'Families of rational numbers with predictable Engle product expansions', *Journal of Integer Sequences*, vol. 12, pp. 1-7.View/Download from: UTS OPUS

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In 1987 Knopfmacher and Knopfmacher published new infinite product expansions for real numbers 0 < A < 1 and A > 1. They called these expansions Engel product expansions. At that time they had difficulty finding rational 0 < A < 1 for which the Engel product expansion is predictable. Later, in 1993, Arnold Knopfmacher presented many such families of rationals. In this paper we add to Arnold Knopfmacher's list of such families.

Melham, R 2009, 'Some conjectures concerning sums of odd powers of Fibonacci and Lucas numbers', *Fibonacci Quarterly*, vol. 46/47, no. 4, pp. 312-315.View/Download from: UTS OPUS

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This paper contains observations, conjectures and open questions concerning two finite sums that involve Fibonacci and Lucas numbers. Certain authors have become aware of the contents of this 9hitherto unpublished) manuscript and have made inroads into some of the challenges it poses. It was felt therefore that te contents of the original manuscript ought tobe made public.

Melham, R 2008, 'Analogues of two classical theorems onthe representations of a number', *Integers: Eletronic Journal of Combinatorial Number Theory 8*, vol. 8, pp. 1-10.View/Download from: UTS OPUS

Melham, R 2008, 'Probable Prime Tests for Generalized Mersenne Numbers', *Boletin: Sociedad Matematica Mexicana*, vol. 14, no. 1, pp. 7-14.View/Download from: UTS OPUS

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The classical Lucas-Lehmer test gives necessary and sufficient conditions for the primality of 2 p -1, p an odd prime. Such primes are called Mersenne primes. Here, taking b = 2 and a = 3 to be integers, and p to be an odd prime, we give probable prime tests for (b p+1)/(b +1) and (a p -1)/(a -1) that are analogous to the classical Lucas-Lehmer test

Melham, RS 2008, 'Some conjectures concerning sums of odd powers of Fibonacci and Lucas numbers', *Fibonacci Quarterly*, vol. 46-47, no. 4, pp. 312-315.

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This paper contains observations, conjectures, and open questions concerning two finite sums that involve Fibonacci and Lucas numbers. Certain authors have become aware of the contents of this (hitherto unpublished) manuscript, and have made inroads into some of the challenges it poses. It was felt, therefore, that the contents of the original manuscript ought to be made public.

Bruckman, PS & Melham, R 2007, 'Some theorems involving powers of generalised Fibonacci numbers at non-equidistant points', *The Fibonacci Quarterly*, vol. 45, no. 3, pp. 208-220.View/Download from: UTS OPUS

Melham, R 2004, 'Certain classes of finite sums that involve generalized Fibonacci and Lucas numbers', *Fibonacci Quarterly*, vol. 42, pp. 47-54.View/Download from: UTS OPUS

Melham, R 2004, 'Ye olde Fibonacci curiosity shoppe revisited', *Fibonacci Quarterly*, vol. 42, pp. 155-160.View/Download from: UTS OPUS

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There are many Fibonacci identities to be found in short informal articles in the early editions of The Fibonacci Quarterly. See, for example, (1) and (2). The aim of the authoprs was to gather Fibonacci identities from diverse sources and display them for all to see. Many of the identities that appeared were quite old and possessed beautiful symmetry. As such, some were alreasy classics, and today appear regularly in research papers. Perhaps the best example is Simson's identity, which has undergone many generalisations.For an up to date account see (7). However, other identities have received little or no attention, and have not deatured in the literature since those early days. The purpose of ths paper is to present some new insight into two such identities. We assume throughout that the sequences in this paper are defined for all integers, and henceforth we do not restate this.

Melham, R 2003, 'A Fibonacci identity in the spirit of Simson and Gelin-Cesaro', *Fibonacci Quarterly*, vol. 41, no. 2, pp. 142-143.

Melham, R 2003, 'A three-variable identity involving cubes of fibonacci numbers', *Fibonacci Quarterly*, vol. 41, no. 3, pp. 220-223.View/Download from: UTS OPUS

Melham, R 2003, 'On some reciprocal sums of Brousseau: An alternative approach to that of Carlitz', *Fibonacci Quarterly*, vol. 41, no. 1, pp. 59-62.View/Download from: UTS OPUS

Melham, R 2002, 'Reduction formulas for the summation of reciprocals in certain secod-order recurring sequences', *Fibonacci Quarterly*, vol. 40, no. 1, pp. 71-75.View/Download from: UTS OPUS

Melham, R 2001, 'Summation of Reciprocals which involve products of terms from generalised Fibonacci sequences - Part II', *The Fibonacci Quarterly*, vol. 39, no. 3, pp. 264-267.View/Download from: UTS OPUS

Melham, R 2000, 'A Generalization of a Result of Andre-Jeannin Concerning Summation of Reciprocals', *Portugaliae Mathematica*, vol. 57, no. 1, pp. 45-47.

Melham, R 2000, 'Alternating sums of Fourth powers of Fibonacci and Lucas Numbers', *The Fibonacci Quarterly*, vol. 38, no. 3, pp. 254-259.

Melham, R 2000, 'On an observation of D'Ocagne concerning the fundamental sequence', *The Fibonacci Quarterly*, vol. 38, no. 5, pp. 446-450.

Melham, R 2000, 'Summation of reciprocals which involve products of terms from Generalized Fibonacci Sequences', *The Fibonacci Quarterly*, vol. 38, no. 4, pp. 294-298.

Melham, R 2000, 'Sums of Certain products of Fibonacci and Lucas numbers - Part II', *The Fibonacci Quarterly*, vol. 38, no. 1.

Melham, R & Cooper, C 2000, 'The Eigenvectors of a certain Matrix of Binomial Coefficients', *The Fibonacci Quarterly*, vol. 38, no. 2, pp. 123-126.

Melham, R 1999, 'A Generalization of a Result of Brillhart', *International journal of Mathematical Education in Science and Technology*, vol. 30, no. 5, pp. 787-787.

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Generalisation of Brillhart's 1964 theorum

Melham, R 1999, 'Families of Identities Involving Sums of Powers of the Fibonacci and Lucas Numbers', *Fibonacci Quarterly*, vol. 37, no. 4, pp. 315-319.

Melham, R 1999, 'Generalizations of Some Identities of Long', *Fibonacci Quarterly*, vol. 37, no. 2, pp. 106-110.View/Download from: UTS OPUS

Melham, R 1999, 'Lambert Series and Elliptic Functions and Certain Reciprocal Sums', *Fibonacci Quarterly*, vol. 37, no. 3, pp. 208-212.View/Download from: UTS OPUS

Melham, R 1999, 'Lucas Sequences and Functions of a 3-by-3 Matrix', *Fibonacci Quarterly*, vol. 37, no. 2, pp. 111-116.View/Download from: UTS OPUS

Melham, R 1999, 'Lucas Sequences and Functions of a 4-by-4 Matrix', *Fibonacci Quarterly*, vol. 37, no. 3, pp. 269-276.View/Download from: UTS OPUS

Melham, R 1999, 'On Sums of Powers of Terms in a Linear Recurrence', *Portugaliae Mathematica*, vol. 56, no. 4, pp. 501-508.View/Download from: UTS OPUS

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ON SUMS OF POWERS OF TERMS IN A LINEAR RECURRENCE

Melham, R 1999, 'Some analogs of the identity F-n(2)+F-n+1(2)=F2n+1', *Fibonacci Quarterly*, vol. 37, no. 4, pp. 305-311.View/Download from: UTS OPUS

Melham, R 1999, 'Sums Involving Fibonacci and Pell Numbers', *Portugaliae Mathematica*, vol. 56, no. 3, pp. 309-317.View/Download from: UTS OPUS

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SUMS INVOLVING FIBONACCI AND PELL NUMBERS

Melham, R 1999, 'Sums of Certain Products of Fibonacci and Lucas Numbers', *Fibonacci Quarterly*, vol. 37, no. 3, pp. 248-251.View/Download from: UTS OPUS

Melham, R 1999, 'Twenty-two Infinite Sums Involving the Sine and Cosine Functions', *International journal of Mathematical Education in Science and Technology*, vol. 30, no. 4, pp. 595-602.View/Download from: UTS OPUS

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Five finite sums derived from the Chebyshev polynomials of the first and second kinds are given. These finite sums, which are shown to satisfy third order linear recurrence relations, are used to derive 22 infinite sums which involve the sine and cosine functions.

Melham, R 1998, 'Generalized Triple Products', *Fibonacci Quarterly*, vol. 36, no. 5, pp. 452-456.View/Download from: UTS OPUS

Melham, R 1997, 'Conics which Characterize Certain Lucas Sequences', *Fibonacci Quarterly*, vol. 35, no. 3, pp. 248-251.View/Download from: UTS OPUS

Shannon, AG & Melham, R 1996, 'Some Aspects of a Partial Difference Equation', *Bulletin of Number Theory and Related Topics*, vol. 16, pp. 31-44.

Melham, R & Jennings, D 1995, 'On the General Linear Recurrence Relation', *Fibonacci Quarterly*, vol. 33, no. 2, pp. 142-146.View/Download from: UTS OPUS

Melham, R & Shannon, AG 1995, 'A Generalization of a Result of D'Ocagne', *Fibonacci Quarterly*, vol. 33, no. 2, pp. 135-138.View/Download from: UTS OPUS

Melham, R & Shannon, AG 1995, 'A Generalization of the Catalan Identity and Some Consequences', *Fibonacci Quarterly*, vol. 33, no. 1, pp. 82-84.View/Download from: UTS OPUS

Melham, R & Shannon, AG 1995, 'Generalizations of Some Simple Congruences', *Fibonacci Quarterly*, vol. 33, no. 2, pp. 126-130.View/Download from: UTS OPUS

Melham, R & Shannon, AG 1995, 'Inverse Trigonometric and Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers', *Fibonacci Quarterly*, vol. 33, no. 1, pp. 32-40.View/Download from: UTS OPUS

Melham, R & Shannon, AG 1995, 'On Reciprocal Sums of Chebyshev Related Sequences', *Fibonacci Quarterly*, vol. 33, no. 3, pp. 194-202.View/Download from: UTS OPUS

Melham, R & Shannon, AG 1995, 'Some Infinite Series Summations Using Power Series Evaluated at a Matrix', *Fibonacci Quarterly*, vol. 33, no. 1, pp. 13-20.View/Download from: UTS OPUS

Melham, R & Shannon, AG 1995, 'Some Summation Identities Using Generalized Q-Matrices', *Fibonacci Quarterly*, vol. 33, no. 1, pp. 64-73.View/Download from: UTS OPUS

Shannon, AG & Melham, R 1995, 'Extended and Generalized Fibonacci Polynomials', *International journal of Mathematical Education in Science and Technology*, vol. 26, no. 2, pp. 296-300.View/Download from: UTS OPUS

Melham, R 1999, 'On Certain Polynomials of Even Subscripted Lucas Numbers', *8th International Research Conference on Fibonacci Numbers and Their Application*, 8th International Research Conference on Fibonacci Numbers and Their Application, Kluwer Academic Publishers, ROCHESTER INST TECHNOL, ROCHESTER, NY, pp. 251-258.View/Download from: UTS OPUS

Melham, R & Shannon, AG 1994, 'On Reciprocal Sums of Second Order Sequences', *The Sixth International Research Conference on Fibonacci Numbers and Their Applications*, The Sixth International Research Conference on Fibonacci Numbers and Their Applications, Kluwer Academic Publishers, Washington State University, Pullman, Washington, USA, pp. 355-364.

Shannon, AG, Loh, RP, Melham, R & Horadam, AN 1994, 'A Search for Solutions of a Functional Equation', *Sixth International Research Conference on Fibonacci Numbers and their Applications*, Sixth International Research Conference on Fibonacci Numbers and their Applications, Kluwer Academic Publishers, Washington State University, Pullman, Washington, U.S.A., pp. 431-441.

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This volume contains the proceedings of the Sixth International Research Conference on Fibonacci Numbers and their Applications. It includes a carefully refereed selection of papers dealing with number patterns, linear recurrences and the application of Fibonacci Numbers to probability, statistics, differential equations, cryptography, computer science and elementary number theory. This volume provides a platform for recent discoveries and encourages further research. It is a continuation of the work presented in the previously published proceedings of the earlier conferences, and shows the growing interest in, and importance of, the pure and applied aspects of Fibonacci Numbers in many different areas of science