Pell Walks

2013 ◽  
Vol 97 (538) ◽  
pp. 27-35 ◽  
Author(s):  
Thomas Koshy
Keyword(s):  
Lattice Paths ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Fertile Ground ◽  
Fibonacci And Lucas Numbers ◽  
Image Position

Like Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers are a fertile ground for creativity and exploration. They also have interesting applications to combinatorics [1], especially to the study of lattice paths [2, 3], as we will see shortly.Pell numbers Pn and Pell-Lucas numbers Qn are often defined recursively [4, 5]:where n ≥ 3. They can also be defined by Binet-like formulas:

Graph-theoretic models for the Fibonacci family

2014 ◽  
Vol 98 (542) ◽  
pp. 256-265 ◽  
Author(s):  
Thomas Koshy
Keyword(s):  
Fibonacci Numbers ◽  
Mathematical Community ◽  
Large Integer ◽  
Lucas Numbers ◽  
Graph Theoretic ◽  
Fertile Ground ◽  
Graph Theoretic Models ◽  
Fibonacci And Lucas Numbers ◽  
Image Position

The well-known Fibonacci and Lucas numbers continue to faxcinate the mathematical community with their beauty, elegance, ubiquity, and applicability. After several centuries of exploration, they are still a fertile ground for additional activities, for Fibonacci enthusiasts and amateurs alike.Fibonacci numbersFnand Lucas numbersLnbelong to a large integer family {xn}, often defined by the recurrencexn=xn−1+xn−2, wherex1=a,x2=b, andn≥ 3. Whena=b= 1,xn=Fn; and whena= 1 andb= 3,xn=Ln. Clearly,F0= 0 andL0= 2. They satisfy a myriad of elegant properties [1,2,3]. Some of them are:In this article, we will give a brief introduction to theQ-matrix, employ it in the construction of graph-theoretic models [4, 5], and then explore some of these identities using them.In 1960 C.H. King studied theQ-matrix

Divisibility properties for Fibonacci and related numbers

2013 ◽  
Vol 97 (540) ◽  
pp. 461-464
Author(s):  
Jawad Sadek ◽  
Russell Euler
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Unified Approach ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Image Position ◽  
Divisibility Properties

Although it is an old one, the fascinating world of Fibonnaci numbers and Lucas numbers continues to provide rich areas of investigation for professional and amateur mathematicians. We revisit divisibility properties for t0hose numbers along with the closely related Pell numbers and Pell-Lucas numbers by providing a unified approach for our investigation.For non-negative integers n, the recurrence relation defined bywith initial conditionscan be used to study the Pell (Pn), Fibonacci (Fn), Lucas (Ln), and Pell-Lucas (Qn) numbers in a unified way. In particular, if a = 0, b = 1 and c = 1, then (1) defines the Fibonacci numbers xn = Fn. If a = 2, b = 1 and c = 1, then xn = Ln. If a = 0, b = 1 and c = 2, then xn = Pn. If a =b = c = 2, then xn = Qn [1].

Vieta-like products of nested radicals

2010 ◽  
Vol 94 (529) ◽  
pp. 62-66
Author(s):  
Thomas J. Osler
Keyword(s):  
Positive Integer ◽  
Recursion Relation ◽  
Golden Section ◽  
Infinite Product ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers ◽  
Image Position ◽  
Analytical Expressions

The beautiful infinite product of radicalsdue to Vieta [1] in 1592, is one of the oldest non-iterative analytical expressions for π, In a previous paper [2] the author proved the following two Vieta-like products:for N even, andfor N odd. Here N is a positive integer, FN and LN are the Fibonacci and Lucas numbers, and is the golden section. (The Fibonacci numbers are F1 = 1, F2 = 1, with the recursion relation , while the Lucas numbers are L1 = 1, L2 = 3 with the same recursion relation )

scholarly journals ON LINEAR RELATIONS FOR DIRICHLET SERIES FORMED BY RECURSIVE SEQUENCES OF SECOND ORDER

2020 ◽  
pp. 1-25
Author(s):  
CARSTEN ELSNER ◽  
NICLAS TECHNAU
Keyword(s):  
Algebraic Independence ◽  
Recurrence Formula ◽  
Zeta Functions ◽  
Second Order ◽  
Lucas Numbers ◽  
Linear Forms ◽  
Recursive Sequences ◽  
Ramanujan Functions ◽  
Fibonacci And Lucas Numbers ◽  
Image Position

Let $F_{n}$ and $L_{n}$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by $$\begin{eqnarray}\unicode[STIX]{x1D701}_{F}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{F}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{L_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{L_{n}^{s}}}.\end{eqnarray}$$ As a consequence of Nesterenko’s proof of the algebraic independence of the three Ramanujan functions $R(\unicode[STIX]{x1D70C}),Q(\unicode[STIX]{x1D70C}),$ and $P(\unicode[STIX]{x1D70C})$ for any algebraic number $\unicode[STIX]{x1D70C}$ with $0<\unicode[STIX]{x1D70C}<1$ , the algebraic independence or dependence of various sets of these numbers is already known for positive even integers $s$ . In this paper, we investigate linear forms in the above zeta functions and determine the dimension of linear spaces spanned by such linear forms. In particular, it is established that for any positive integer $m$ the solutions of $$\begin{eqnarray}\mathop{\sum }_{s=1}^{m}(t_{s}\unicode[STIX]{x1D701}_{F}(2s)+u_{s}\unicode[STIX]{x1D701}_{F}^{\ast }(2s)+v_{s}\unicode[STIX]{x1D701}_{L}(2s)+w_{s}\unicode[STIX]{x1D701}_{L}^{\ast }(2s))=0\end{eqnarray}$$ with $t_{s},u_{s},v_{s},w_{s}\in \mathbb{Q}$ $(1\leq s\leq m)$ form a $\mathbb{Q}$ -vector space of dimension $m$ . This proves a conjecture from the Ph.D. thesis of Stein, who, in 2012, was inspired by the relation $-2\unicode[STIX]{x1D701}_{F}(2)+\unicode[STIX]{x1D701}_{F}^{\ast }(2)+5\unicode[STIX]{x1D701}_{L}^{\ast }(2)=0$ . All the results are also true for zeta functions in $2s$ , where the Fibonacci and Lucas numbers are replaced by numbers from sequences satisfying a second-order recurrence formula.

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

BASIS FOR SYNTHESIS OF SPIRAL LATTICE QUASICRYSTALS

1989 ◽  
Vol 03 (14) ◽  
pp. 1071-1085 ◽  
Author(s):  
L. A. BURSILL ◽  
GEORGE RYAN ◽  
XUDONG FAN ◽  
J. L. ROUSE ◽  
JULIN PENG ◽  
...  
Keyword(s):  
Experimental Error ◽  
Theoretical Models ◽  
Cell Number ◽  
Close Packing ◽  
Helianthus Tuberosus ◽  
Lucas Numbers ◽  
Average Growth ◽  
Left And Right ◽  
Seed Cell ◽  
Fibonacci And Lucas Numbers

Observations of the sunflower Helianthus tuberosus reveal the occurrence of both Fibonacci and Lucas numbers of visible spirals (parastichies). This species is multi-headed, allowing a quantitative study of the relative abundance of these two types of phyllotaxis. The florets follow a spiral arrangement. It is remarkable that the Lucas series occurred, almost invariably, in the first-flowering heads of individual plants. The occurrence of left-and right-handed chirality was found to be random, within experimental error, using an appropriate chirality convention. Quantitative crystallographic studies allow the average growth law to be derived (r = alτ−1; θ = 2πl/(τ + 1), where a is a constant, l is the seed cell number and τ is the golden mean [Formula: see text]). They also reveal departures from classical theoretical models of phyllotaxis, taking the form of persistent oscillations in both divergence angle and radius. The experimental results are discussed in terms of a new theoretical model for the close-packing of growing discs. Finally, a basis for synthesis of (inorganic) spiral lattice structures is proposed.

Closed form evaluation of sums containing squares of Fibonomial coefficients

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Closed Form ◽  
Systematic Approach ◽  
Sums Of Squares ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Finite Products ◽  
Fibonacci And Lucas Numbers ◽  
Form Evaluation

AbstractWe give a systematic approach to compute certain sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. The technique is to rewrite everything in terms of a variable

scholarly journals Identities on generalized Fibonacci and Lucas numbers

2020 ◽  
Vol 26 (3) ◽  
pp. 189-202
Author(s):  
K. M. Nagaraja ◽  
◽  
P. Dhanya ◽  
Keyword(s):  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers
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