Evaluation of sums containing triple aerated generalized Fibonomial coefficients

2017 ◽  
Vol 67 (2) ◽  
Author(s):  
Emrah Kiliç
Keyword(s):  
Fibonacci Number ◽  
Fibonomial Coefficients

AbstractWe evaluate a class of sums of triple aerated Fibonomial coefficients with a generalized Fibonacci number as coefficient. The technique is to rewrite everything in terms of a variable

On some new identities for the Fibonomial coefficients

2014 ◽  
Vol 64 (4) ◽  
Author(s):  
Diego Marques ◽  
Pavel Trojovský
Keyword(s):  
Fibonacci Number ◽  
Fibonomial Coefficients

AbstractLet F n be the nth Fibonacci number. The Fibonomial coefficients $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F$$ are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with $$\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1$$ and $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0$$. In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l.

scholarly journals On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 962
Author(s):  
Eva Trojovská  ◽  
Pavel Trojovský
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Initial Values

Let (tn(r))n≥0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence tn(r)=tn−1(r)+⋯+tn−r(r) for n≥r, with initial values t0(r)=0 and ti(r)=1, for all 1≤i≤r. In 2002, Grossman and Luca searched for terms of the sequence (tn(2))n, which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ≥1, there exists an effectively computable constant C=C(ℓ)>0 (only depending on ℓ), such that, if (m,n,r) is a solution of tm(r)=n!+(n+1)!+⋯+(n+ℓ)!, with r even, then max{m,n,r}<C. As an application, we solve the previous equation for all 1≤ℓ≤5.

Fibonacci Number Triples

1961 ◽  
Vol 68 (8) ◽  
pp. 751 ◽  
Author(s):  
A. F. Horadam
Keyword(s):  
Fibonacci Number

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 The Tight Upper Bound for the Number of Matchings of Tricyclic Graphs

10.37236/2165 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Ardeshir Dolati ◽  
Somayyeh Golalizadeh
Keyword(s):  
Upper Bound ◽  
Fibonacci Number ◽  
Tricyclic Graph ◽  
Tricyclic Graphs

In this paper, we determine the tight upper bound for the number of matchings of connected $n$-vertex tricyclic graphs. We show that this bound is $13 f_{n-4} + 16f_{n-5}$, where $f_n$ be the  $n$th Fibonacci number. We also  characterize the $n$-vertex simple connected tricyclic graph for which the bound is best  possible.A corrigendum was added to this paper on Jun 17, 2015. 

scholarly journals DMRG investigation of constrained models: from quantum dimer and quantum loop ladders to hard-boson and Fibonacci anyon chains

2019 ◽  
Vol 6 (3) ◽  
Author(s):  
Natalia Chepiga ◽  
Frédéric Mila
Keyword(s):  
Model Building ◽  
Fibonacci Number ◽  
Complete Analysis ◽  
Loop Model ◽  
Boundary Field ◽  
Conformal Field ◽  
Density Matrix Renormalization ◽  
Early Results ◽  
Boson Model ◽  
Constrained Models

Motivated by the presence of Ising transitions that take place entirely in the singlet sector of frustrated spin-1/2 ladders and spin-1 chains, we study two types of effective dimer models on ladders, a quantum dimer model and a quantum loop model. Building on the constraints imposed on the dimers, we develop a Density Matrix Renormalization Group algorithm that takes full advantage of the relatively small Hilbert space that only grows as Fibonacci number. We further show that both models can be mapped rigorously onto a hard-boson model first studied by Fendley, Sengupta and Sachdev [Phys. Rev. B 69, 075106 (2004)], and combining early results with recent results obtained with the present algorithm on this hard-boson model, we discuss the full phase diagram of these quantum dimer and quantum loop models, with special emphasis on the phase transitions. In particular, using conformal field theory, we fully characterize the Ising transition and the tricritical Ising end point, with a complete analysis of the boundary-field correspondence for the tricritical Ising point including partially polarized edges. Finally, we show that the Fibonacci anyon chain is exactly equivalent to special critical points of these models.

Spiral growth inNephrolepidina: evidence of “golden selection”

Paleobiology ◽  
10.1666/12057 ◽  
2014 ◽  
Vol 40 (2) ◽  
pp. 151-161 ◽  
Author(s):  
Andrea Benedetti
Keyword(s):  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Spiral Growth ◽  
Regular Growth ◽  
Mean Values ◽  
Evolutionary Selection ◽  
Golden Angle ◽  
New Type ◽  
New Evidence ◽  
Interpretative Model

Examination of the neanic apparatuses of known populations ofNephrolepidina praemarginata,N. morgani, andN. tournouerireveals that the equatorial chamberlets are arranged in spirals, along the direction of connection of the oblique stolons, giving the optical effect of intersecting curves. InN. praemarginatacommonly 34 left- and right-oriented primary spirals occur from the first annulus to the periphery, 21 secondary spirals from the third to fifth annulus, 13 ternary spirals from the fifth to eighth annulus, following the Fibonacci sequence.The number of the spirals increases in larger specimens and in more embracing morphotypes, and especially in trybliolepidine specimens; the secondary and ternary spirals from the investigatedN. praemarginatatoN. tournoueripopulations tend to start from more distal annuli. An interpretative model of the spiral growth ofNephrolepidinais attempted.The angle formed by the basal annular stolon and distal oblique stolon in equatorial chamberlets ranges from 122° inN. praemarginatato mean values close to the golden angle (137.5°) inN. tournoueri.The increase in the Fibonacci number of spirals during the evolution of the lineage, along with the disposition of the stolons between contiguous equatorial chamberlets, provides new evidence of evolutionary selection for specimens with optimally packed chamberlets.Natural selection favors individuals with the most regular growth, which fills the equatorial space more efficiently, thus allowing these individuals to reach the adult stage faster. We refer to this new type of selection as “golden selection.”

scholarly journals Fibonacci Numbers with a Prescribed Block of Digits

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 639 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Linear Forms

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

An energy-efficient wireless communication scheme using Quint Fibonacci number system

2016 ◽  
Vol 16 (2) ◽  
pp. 140 ◽  
Author(s):  
Ansuman Bhattacharya ◽  
Pratham Majumder ◽  
Koushik Sinha ◽  
Bhabani P. Sinha ◽  
K.V.N. Kavitha
Keyword(s):  
Wireless Communication ◽  
Energy Efficient ◽  
Fibonacci Number ◽  
Number System ◽  
Communication Scheme
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