More Power to Fibonacci

2003 ◽  
Vol 87 (509) ◽  
pp. 196-198 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Linear Recurrence

You should never judge a book by its title - but I nearly always do. A mathematical colleague recommended a book to me Concrete Mathematics [1]. Just recently, someone else also referred me to the same book. Well a wink’s as good as a nod to a blind man and I now realise what a mistake I’d made; concrete is a conflation of ‘continuous’ and ‘discrete’. Like all good textbooks in mathematics, it distinguishes itself not just by the quality of its contents and presentation, but also by the superb collection of problems that it includes. One such problem - question 58 in Chapter 6 - inspired this article. There was a hint towards a generalised result, but there were no details. It did give a reference, [2, 3], but despite some efforts I found this unobtainable. Hence this article and my attempt to cover the same ground. The question posed in the book is concerned with the linear recurrence relations satisfied by the successive powers of Fibonacci numbers - so we start with the Fibonacci sequence itself.

scholarly journals Extended Fibonacci numbers and polynomials with probability applications

2004 ◽  
Vol 2004 (50) ◽  
pp. 2681-2693
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Generating Function ◽  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Multinomial Coefficients

The extended Fibonacci sequence of numbers and polynomials is introduced and studied. The generating function, recurrence relations, an expansion in terms of multinomial coefficients, and several properties of the extended Fibonacci numbers and polynomials are obtained. Interesting relations between them and probability problems which take into account lengths of success and failure runs are also established.

Delving Deeper: Simple Recurrence Relations, Proper Guessing, and Closed-Form Solutions

2005 ◽  
Vol 99 (4) ◽  
pp. 292-295
Author(s):  
M. J. Nandor
Keyword(s):  
Closed Form ◽  
Recurrence Relations ◽  
Functional Form ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
School Level ◽  
Discrete Mathematics ◽  
High School Level ◽  
Closed Form Solutions ◽  
Mathematics Class

In reading the February 2004 issue of the Mathematics Teacher, I found the wonderful “Delving Deeper” article “Fibonacci and Related Sequences,” by Richard Askey. At the end, in the editors' note, the question was posed whether Fibonacci numbers can be found in a closed form. As it happened, I was teaching that exact topic in my discrete mathematics class at the time. Ever since taking discrete mathematics in college, recurrence relations have fascinated me, and the derivation of a functional form of the Fibonacci sequence is just one great example of how recurrence relations can be used, even at the high school level.

How the odd terms in the Fibonacci sequence stack up

2006 ◽  
Vol 90 (519) ◽  
pp. 431-442 ◽  
Author(s):  
S. Rinaldi ◽  
D. G. Rogers
Keyword(s):  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence

Dedicated to H. N. V. Temperley on the occasion of his ninetieth birthday 4 March 2005The authors of the recent Note [1] exhibit an odd preference. They derive recurrence relations for the odd terms, un = F2n+1, n ≥ 0, in the sequence of Fibonacci numbers, Fn, defined by

scholarly journals Individual Gap Measures from Generalized Zeckendorf Degompositions

2017 ◽  
Vol 12 (1) ◽  
pp. 27-36 ◽  
Author(s):  
Robert Dorward ◽  
Pari L. Ford ◽  
Eva Fourakis ◽  
Pamela E. Harris ◽  
Steven. J. Miller ◽  
...  
Keyword(s):  
Positive Integer ◽  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Linear Recurrence ◽  
The Individual ◽  
Geometric Decay ◽  
Almost All ◽  
Zeckendorf’S Theorem ◽  
General Conditions

Abstract Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gajw between summands for m € [Fn,Fn+1) converge to geometric decay for almost all m as n→ ∞. While similar results are known for many other recurrences, previous work focused on proving Gaussianity for the number of summands or the average gap measure. We derive general conditions, which are easily checked, that yield geometric decay in the individual gap measures of generalized Zerkendorf decompositions attached to many linear recurrence relations.

scholarly journals On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

2016 ◽  
Vol 67 (1) ◽  
pp. 41-46
Author(s):  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Initial Values ◽  
Lucas Sequence ◽  
Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

scholarly journals On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Σn k=0 kW3 k and Σn k=1 kW3− k

2020 ◽  
pp. 37-52 ◽  
Author(s):  
Yuksel Soykan
Keyword(s):  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Special Cases

In this paper, closed forms of the sum formulas Σn k=0 kW3 k and Σn k=1 kW3-k for the cubes of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Our work generalize second order recurrence relations.

scholarly journals On Fibonacci Numbers and its Applications

2019 ◽  
Vol 8 (12) ◽  
pp. 453-455
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence

In this article, we explore the representation of the product of k consecutive Fibonacci numbers as the sum of kth power of Fibonacci numbers. We also present a formula for finding the coefficients of the Fibonacci numbers appearing in this representation. Finally, we extend the idea to the case of generalized Fibonacci sequence and also, we produce another formula for finding the coefficients of Fibonacci numbers appearing in the representation of three consecutive Fibonacci numbers as a particular case. Also, we point out some amazing applications of Fibonacci numbers.

scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

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