Closed form evaluation of sums containing squares of Fibonomial coefficients

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

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On sums of squares of Fibonomial coefficients by q-calculus

Asian-European Journal of Mathematics ◽

10.1142/s1793557116500625 ◽

2016 ◽

Vol 09 (03) ◽

pp. 1650062

Author(s):

Emrah Kılıç ◽

Aynur Yalçıner

Keyword(s):

Closed Form ◽

Generating Functions ◽

Sums Of Squares ◽

Classical Formula ◽

Lucas Numbers ◽

Fibonomial Coefficients ◽

Finite Products ◽

Fibonacci And Lucas Numbers ◽

Form Evaluation

We present some new kinds of sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. As proof method, we will follow the method given in [E. Kılıç and H. Prodinger, Closed form evaluation of sums containing squares of Fibonomial coefficients, accepted in Math. Slovaca]. For this, first we translate everything into [Formula: see text]-notation, and then to use generating functions and Rothe’s identity from classical [Formula: see text]-calculus.

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

Mathematica Slovaca ◽

10.1515/ms-2017-0378 ◽

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.

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BASIS FOR SYNTHESIS OF SPIRAL LATTICE QUASICRYSTALS

Modern Physics Letters B ◽

10.1142/s0217984989001667 ◽

1989 ◽

Vol 03 (14) ◽

pp. 1071-1085 ◽

Cited By ~ 1

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.

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