scholarly journals Some Finite Sums Involving Generalized Fibonacci and Lucas Numbers

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
E. Kılıç ◽  
N. Ömür ◽  
Y. T. Ulutaş
Keyword(s):  
Lucas Numbers ◽  
Alternating Sums ◽  
Finite Sums ◽  
Fibonacci And Lucas Numbers

By considering Melham's sums (Melham, 2004), we compute various more general nonalternating sums, alternating sums, and sums that alternate according to involving the generalized Fibonacci and Lucas numbers.

scholarly journals Alternating sums of the powers of Fibonacci and Lucas numbers

2011 ◽  
Vol 12 (1) ◽  
pp. 87 ◽  
Author(s):  
Emrah Kiliç ◽  
Neşe Ömür ◽  
Yücel Türker Ulutaş
Keyword(s):  
Lucas Numbers ◽  
Alternating Sums ◽  
Fibonacci And Lucas Numbers

scholarly journals Sums of products of generalized Fibonacci and Lucas numbers

2009 ◽  
Vol 42 (2) ◽  
Author(s):  
Zvonko Čerin
Keyword(s):  
Recent Work ◽  
Recurrence Relation ◽  
Fixed Number ◽  
Binomial Coefficients ◽  
Natural Numbers ◽  
Lucas Numbers ◽  
Alternating Sums ◽  
Explicit Formulae ◽  
Fibonacci And Lucas Numbers ◽  
Sums Of Products

AbstractIn this paper we obtain explicit formulae for sums of products of a fixed number of consecutive generalized Fibonacci and Lucas numbers. These formulae are related to the recent work of Belbachir and Bencherif. We eliminate all restrictions about the initial values and the form of the recurrence relation. In fact, we consider six different groups of three sums that include alternating sums and sums in which terms are multiplied by binomial coefficients and by natural numbers. The proofs are direct and use the formula for the sum of the geometric series.

scholarly journals Finite Sums Involving Reciprocals of the Binomial and Central Binomial Coefficients and Harmonic Numbers

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2002
Author(s):  
Necdet Batir ◽  
Anthony Sofo
Keyword(s):  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Lucas Numbers ◽  
Finite Sums ◽  
Fibonacci And Lucas Numbers ◽  
Finite Sum

We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known results, while the others are new.

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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