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Journal articles on the topic 'Fibonacci theory'

Author: Grafiati

Published: 28 May 2022

Last updated: 30 July 2025

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1

Şentürk, Gülsüm Yeliz, and Nurten Gürses. "Dual quaternion theory over HGC numbers." Journal of Discrete Mathematical Sciences & Cryptography 27, no. 1 (2024): 117–42. http://dx.doi.org/10.47974/jdmsc-1611.

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Knowing the applications of quaternions in various fields, such as robotics, navigation, computer visualization and animation, in this study, we give the theory of dual quaternions considering Hyperbolic-Generalized Complex (HGC) numbers as coefficients via generalized complex and hyperbolic numbers. We account for how HGC number theory can extend dual quaternions to HGC dual quaternions. Some related theoretical results with HGC Fibonacci/Lucas numbers are established, including their dual quaternions. Given HGC Fibonacci/Lucas numbers, their special matrix correspondences have been identifie

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2

Anderson, Peter G. "Extended Fibonacci Zeckendorf Theory." Fibonacci Quarterly 52, no. 5 (2014): 15–21. http://dx.doi.org/10.1080/00150517.2014.12427852.

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3

BASU, MANJUSRI, and MONOJIT DAS. "CODING THEORY ON FIBONACCI n-STEP NUMBERS." Discrete Mathematics, Algorithms and Applications 06, no. 02 (2014): 1450017. http://dx.doi.org/10.1142/s1793830914500177.

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We consider the series of Fibonacci n-step numbers and a class of square matrix of order n based on Fibonacci n-step numbers with determinant +1 or -1. Thereby, we introduce a new coding theory called Fibonacci n-step coding theory and establish generalized relation among the code elements for all values of n. For n = 2, the correct ability of the method is 93.33%; for n = 3, the correct ability of the method is 99.80% and for n = 4, the correct ability of the method is 99.998%. In general, the correct ability of the method increases as n increases.

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4

BASU, MANJUSRI, and BANDHU PRASAD. "CODING THEORY ON h(x) FIBONACCI p-NUMBERS POLYNOMIALS." Discrete Mathematics, Algorithms and Applications 04, no. 03 (2012): 1250030. http://dx.doi.org/10.1142/s1793830912500309.

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In this paper, we introduce h(x) Fibonacci p-numbers polynomials and a new Fibonacci Qp,h(x) matrix for h(x) Fibonacci p-numbers polynomials where p(> 0) is integer and h(x)(> 0) is a polynomial with real coefficients. Thereby, we discuss various properties of Qp,h(x) matrix and the coding theory followed from the Qp,h(x) matrix. We establish the relations among the code matrix elements for p(> 0) is integer and h(x)(> 0). We also show that the relations, among the code matrix elements for h(x) = m (> 0), coincide with the relations among the code matrix elements for coding theo

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5

BASU, MANJUSRI, and BANDHU PRASAD. "CODING THEORY ON THE (m, t)-EXTENSION OF THE FIBONACCI p-NUMBERS." Discrete Mathematics, Algorithms and Applications 03, no. 02 (2011): 259–67. http://dx.doi.org/10.1142/s1793830911001097.

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In this paper, we define (m, t)-extension of the Fibonacci p-numbers and Golden (p, m, t)-proportions, where p ≥ 0 is integer, m > 0, and t > 0. We establish a relation among golden (p, m, t)-proportion, golden (p, m)-proportion and golden p-proportion. Thereby, we define a new Fibonacci Gp, m, t matrix. Then we show that by proper selection of the initial terms for the (m, t)-extension of the Fibonacci p-numbers, we can apply Fibonacci coding/decoding in Gp, m, t matrix. Also it is obvious that for t = 1, the relations among the code elements for all values of p (non-negative integer) a

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6

Kumar, Nand Kishor, and Suresh Kumar Sahani. "Generalized K-Fibonacci Sequence of Q- Matrix." Mikailalsys Journal of Advanced Engineering International 2, no. 1 (2025): 55–62. https://doi.org/10.58578/mjaei.v2i1.4811.

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This article explores the generalized K-Fibonacci sequence derived from matrices, extending the classical Fibonacci sequence to matrix representations. The properties and its applications are also described. The interplay of matrix algebra and generalized Fibonacci sequences offers insights into advanced sequence theory.

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Mehraban, Elahe, and Mansour Hashemi. "An Application of the t -Extension of the p -Fibonacci Pascal Matrix in Coding Theory." Advances in Mathematical Physics 2022 (October 10, 2022): 1–14. http://dx.doi.org/10.1155/2022/4619136.

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We consider the t -extension of the p -Fibonacci Pascal matrix. First, we study the k -th power of the t -extension of the p -Fibonacci lower and upper triangular Pascal matrix. Then, we obtain a new code which is named the t -extension of the p -Fibonacci Pascal matrix coding/decoding by using them.

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8

Balouria, Pallavi, та Meenakshi Chawla. "On λ-Fibonacci statistical convergence of order- α in intuitionistic fuzzy normed spaces (IFNS)". Journal of Interdisciplinary Mathematics 27, № 8 (2024): 1827–35. https://doi.org/10.47974/jim-2047.

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The main goal of this paper is to study and develop the theory of λ-Fibonacci statisticalconvergence of order-α in IFNS. Also λ-Fibonacci statistical Cauchy sequence of order-α and λ-Fibonacci statistical completeness of order-α in IFNS have been defined and studied.

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RIKLUND, ROLF, MATTIAS SEVERIN, and YOUYAN LIU. "THE THUE-MORSE APERIODIC CRYSTAL, A LINK BETWEEN THE FIBONACCI QUASICRYSTAL AND THE PERIODIC CRYSTAL." International Journal of Modern Physics B 01, no. 01 (1987): 121–32. http://dx.doi.org/10.1142/s0217979287000104.

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The electronic spectrum and eigenstates of a one-dimensional aperiodic Thue-Morse crystal isstudied with an on-site tight-binding model. The relation between the constructing elements andthe hierarchical splitting of the bands into subbands is analysed. The eigenstates are shown to be much more similar to those of a periodic crystal than those of a Fibonacci quasicrystal. We thus claim that the Thue-Morse aperiodic crystal is a link between the Fibonacci quasicrystal and theperiodic crystal, and that the study of non-Fibonaccian aperiodic crystals is a promising steptowards the desired unified

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10

Sharma, Sonal, та Meenakshi Chawla. "Fibonacci statistical convergence of double sequences of order α– in intuitionistic fuzzy normed spaces (IFNS)". Journal of Interdisciplinary Mathematics 27, № 8 (2024): 1817–26. https://doi.org/10.47974/jim-2046.

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The main objective of this work is to study and develop the theory of Fibonacci statistical convergence of double sequences of order α– in IFNS. We examine the Fibonacci statistically Cauchy double sequences of order α– and Fibonacci statistically completeness of double sequences of order α– in IFNS.

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11

Yurttas Gunes, Aysun, Sadik Delen, Musa Demirci, Ahmet Sinan Cevik, and Ismail Naci Cangul. "Fibonacci Graphs." Symmetry 12, no. 9 (2020): 1383. http://dx.doi.org/10.3390/sym12091383.

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Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the golden ratio and also due to many applications in different areas from Biology, Architecture, Anatomy to Finance. In this paper, we define Fibonacci graphs as graphs having degree sequence consisting of n consecutive Fibonacci numbers and use the invariant Ω to obtain some more information on these graphs. We give the necessary and sufficient conditi

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12

Prasad, Bandhu. "The generalized relations among the code elements for a new complex Fibonacci matrix." Discrete Mathematics, Algorithms and Applications 11, no. 02 (2019): 1950026. http://dx.doi.org/10.1142/s1793830919500265.

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In this paper, we introduce a new complex Fibonacci matrix [Formula: see text] whose elements are complex Fibonacci numbers and we developed a new coding and decoding method followed from this complex Fibonacci matrix [Formula: see text]. We establish the relations among the code matrix elements, error detection and correction for this coding theory.

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13

Luca, Florian, Carl Pomerance, and Stephan Wagner. "Fibonacci integers." Journal of Number Theory 131, no. 3 (2011): 440–57. http://dx.doi.org/10.1016/j.jnt.2010.09.010.

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14

Et. al., Vipin Verma,. "Multiplicative Triple Fibonacci Sequence of Third Order." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 11 (2021): 451–58. http://dx.doi.org/10.17762/turcomat.v12i11.5907.

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The Coupled Fibonacci Sequence are Firstly established by K.T. Atanassov in 1985. The abstractions of Fibonacci Triple Sequence are considered in 1987. Fibonacci Sequence stands as a kind of super sequence with Fabulous properties. This is the explosive development in the region of Fibonacci Sequence. Fibonacci was advanced by Leonardo de Pisa (whose nickname was Fibonacci, which means son of Bonacci) in 1202 as a result of his inspection on the growth of a population of rabbits. The consecutive Fibonacci numbers are produced by adding together the two previous numbers in the sequence, after s

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15

Kumar, Nand Kishor, and Suresh Kumar Sahani. "Matrices of Fibonacci Numbers." Mikailalsys Journal of Mathematics and Statistics 3, no. 1 (2024): 71–80. https://doi.org/10.58578/mjms.v3i1.4398.

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This paper describes the matrix representation of Fibonacci numbers. The interaction between number theory and linear algebra is emphasized by the study of Fibonacci numbers using matrices. This viewpoint not only makes calculation easier, but it also reveals the sequence's underlying structural characteristics.

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16

KAYMAK, Öznur. "Coding theory for h(x)-Fibonacci polynomials." Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 26, no. 1 (2024): 226–36. http://dx.doi.org/10.25092/baunfbed.1347379.

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The amount of information transmitted over the internet network has dramatically increased with the prevailing of internet use. As a result of this increase, the algorithms used in data encryption methods have gained importance. In this paper, h(x)-Fibonacci coding/decoding method for h(x)-Fibonacci polynomials is introduced. The proposed method is fast because it is based on basic matrix operations, and it is suitable for cryptographic applications because it uses the ASCII character encoding system. For this reason, it differs from the classical algebraic methods in literature. Furthermore,

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17

HAN, JEONG SOON, HEE SIK KIM, and J. NEGGERS. "THE FIBONACCI-NORM OF A POSITIVE INTEGER: OBSERVATIONS AND CONJECTURES." International Journal of Number Theory 06, no. 02 (2010): 371–85. http://dx.doi.org/10.1142/s1793042110003009.

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In this paper, we define the Fibonacci-norm [Formula: see text] of a natural number n to be the smallest integer k such that n|Fk, the kth Fibonacci number. We show that [Formula: see text] for m ≥ 5. Thus by analogy we say that a natural number n ≥ 5 is a local-Fibonacci-number whenever [Formula: see text]. We offer several conjectures concerning the distribution of local-Fibonacci-numbers. We show that [Formula: see text], where [Formula: see text] provided Fm+k ≡ Fm (mod n) for all natural numbers m, with k ≥ 1 the smallest natural number for which this is true.

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18

Matoušová, Ivana, and Pavel Trojovský. "On Coding by (2,q)-Distance Fibonacci Numbers." Mathematics 8, no. 11 (2020): 2058. http://dx.doi.org/10.3390/math8112058.

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In 2006, A. Stakhov introduced a new coding/decoding process based on generating matrices of the Fibonacci p-numbers, which he called the Fibonacci coding/decoding method. Stakhov’s papers have motivated many other scientists to seek certain generalizations by introducing new additional coefficients into recurrence of Fibonacci p-numbers. In 2013, I. Włoch et al. studied (2,q)-distance Fibonacci numbers F2(q,n) and found some of their combinatorial properties. In this paper, we state a new coding theory based on the sequence (Tq(n))n=−∞∞, which is an extension of Włoch’s sequence (F2(q,n))n=0∞

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19

Chen, Zhaoxue. "Researches on Mathematical Relationship of Five Elements of Containing Notes and Fibonacci Sequence Modulo 5." Scientific World Journal 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/189357.

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Considering the five periods and six qi’s theory in TCM almost shares a common basis of stem-branch system with the five elements of containing notes, studying the principle or mathematical structure behind the five elements of containing notes can surely bring a novel view for the five periods and six qi’s researches. By analyzing typical mathematical rules included in He tu, Luo shu, and stem-branch theory in TCM as well as the Fibonacci sequence especially widely existent in the biological world, novel researches are performed on mathematical relationship between the five elements of contai

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20

Shookooh, Besharat Rezaei, Alireza Monajati, and Hamid Khodabakhshi. "Theory, Design, and Implementation of a New Family of Ultra-Wideband Metamaterial Microstrip Array Antennas Based on Fractal and Fibonacci Geometric Patterns." Journal of Electromagnetic Engineering and Science 20, no. 1 (2020): 53–63. http://dx.doi.org/10.26866/jees.2020.20.1.53.

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The theory and design of a new family of ultra-wideband (UWB) metamaterial (MTM) microstrip array antennas based on fractal and Fibonacci geometric patterns are investigated. First, the UWB microstrip array antenna is presented with two radiating MTM elements. Then, using fractal and Fibonacci geometric patterns, the array antenna is expanded. Improvements in the antenna parameters is achieved by repeating the second and third iterations of the fractal and Fibonacci patterns. As the order of iteration of the fractal and Fibonacci geometric patterns increases, the impedance bandwidth of the MTM

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21

Kürüz, Ferhat. "Some New Fibonacci Matrices and Error Detecting-Correcting Codes." Symmetry 17, no. 3 (2025): 333. https://doi.org/10.3390/sym17030333.

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The rapid and accurate transfer of information is essential in today’s digital world, where error detecting and correcting codes play a crucial role. In this context, Fibonacci numbers have been extensively applied in coding theory, particularly in coding structures based on Fibonacci polynomials and symmetric matrices. This study proposes a novel coding framework by defining and using some new Fibonacci matrices with their determinants. The properties of these matrices are analyzed to construct efficient coding and error correction mechanisms. The proposed structure enables the transmission o

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22

Abraham, Tessymol, and Shiny Jose. "ENERGY OF FIBONACCI PRODUCT CORDIAL GRAPH." South East Asian Journal of Mathematics and Mathematical Sciences 19, no. 02 (2023): 223–30. http://dx.doi.org/10.56827/seajmms.2023.1902.16.

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Let G be a Fibonacci product cordial graph with n vertices. Then the Fibonacci product cordial label energy of G is denoted by EFPCL [G] and is defined as EFPCL[G]= n i=1 |ρi|, where ρi are the eigenvalues of the Fibonacci product cordial labeled matrix. In this paper, we introduce the Fibonacci product cordial labeled matrix, Fibonacci Product Cordial Labeled Laplacian matrix, Fibonacci product cordial label Laplacian energy, Fibonacci product cordial label equi-laplacian energic graphs respectively.

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23

Chow, Sam, and Tom Slattery. "On Fibonacci partitions." Journal of Number Theory 225 (August 2021): 310–26. http://dx.doi.org/10.1016/j.jnt.2021.02.010.

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24

Chung, Fan, Ron Graham, and Sam Spiro. "Slow Fibonacci walks." Journal of Number Theory 210 (May 2020): 142–70. http://dx.doi.org/10.1016/j.jnt.2019.09.016.

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25

Cilleruelo, Javier, and Florian Luca. "Fibonacci lattice points." Ramanujan Journal 22, no. 3 (2010): 285–92. http://dx.doi.org/10.1007/s11139-010-9226-5.

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26

Sikhwal, Omprakash, and Ashish Rastogi. "Domination, Independence and Fibonacci Numbers in Graphs Containing Disjoint Cycles." International Journal of Computational and Applied Mathematics & Computer Science 2 (September 2, 2022): 65–68. http://dx.doi.org/10.37394/232028.2022.2.12.

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Graph theory is a delightful playground for the exploration of proof techniques in discrete mathematics and its results have applications in many areas of the computing, social, and natural sciences. The fastest growing area within graph theory is the study of domination and Independence numbers. Domination number is the cardinality of a minimum dominating set of a graph. Independence number is the maximal cardinality of an independent set of vertices of a graph. The concept of Fibonacci numbers of graphs was first introduced by Prodinger and Tichy in 1982. The Fibonacci numbers of a graph is

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27

Alsoboh, Abdullah, Ala Amourah, Omar Alnajar, Mamoon Ahmed, and Tamer M. Seoudy. "Exploring q-Fibonacci Numbers in Geometric Function Theory: Univalence and Shell-like Starlike Curves." Mathematics 13, no. 8 (2025): 1294. https://doi.org/10.3390/math13081294.

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Emphasising their connection with shell-like star-like curves, this work investigates a new subclass of star-like functions defined by q-Fibonacci numbers and q-polynomials. We study the geometric and analytic properties of this subclass, including the computation of intervals of univalence and nonunivalence for some functions. Moreover, we define a sufficient condition for functions in this subclass to satisfy the criteria of the famous class of analytic functions with positive real components. This work improves our understanding of the link between Fibonacci-type sequences and the geometric

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28

Domada, Triveni, S. Ashok Kumar, Gudela Ashok, and D. Chaya Kumari. "A novel multiphase encryption strategy with fibonacci numbers and matrices." Journal of Discrete Mathematical Sciences and Cryptography 28, no. 1 (2025): 117–29. https://doi.org/10.47974/jdmsc-2069.

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This paper proposes multiple encryption methods that secures plaintext by integrating various techniques, including the application of graph theory to trees, Fibonacci matrices, and affine transformations. The use of multiple encryption layers minimizes the risks associated with data encryption by ensuring that the compromise of a single layer does not jeopardize the overall security. This multiencryption method can be extended to public key cryptosystems. We introduce a super encryption technique that employs Laplace transformations and Fibonacci numbers. The process begins by applying the La

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Kumari, Munesh, Kalika Parasd, and Jagmohan Tanti Tanti. "A NOTE ON LINEAR CODES WITH GENERALIZED FIBONACCI MATRICES." jnanabha 52, no. 02 (2022): 77–81. http://dx.doi.org/10.58250/jnanabha.2022.52209.

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In this paper, we investigate the linear codes from generalized Fibonacci matrices in the context of coding theory. We show that Fibonacci matrices form a generator matrix for the first order ReedMuller codes R(1, 1). Further, we see that Multinacci matrices form a basis for [n, n, 1] MDS-code.

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Prasad, Bandhu. "High rates of Fibonacci polynomials coding theory." Discrete Mathematics, Algorithms and Applications 06, no. 04 (2014): 1450053. http://dx.doi.org/10.1142/s1793830914500530.

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In this paper, a new class of square matrix [Formula: see text] of order pm is introduced, where (p = 3, 4, 5, …), (m = 1, 2, 3, …) and for integers n, x ≥ 1. Fibonacci polynomial coding and decoding methods are followed from [Formula: see text] matrix and high code rates are obtained.

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31

Bilyk, Dmitriy, V. N. Temlyakov, and Rui Yu. "Fibonacci sets and symmetrization in discrepancy theory." Journal of Complexity 28, no. 1 (2012): 18–36. http://dx.doi.org/10.1016/j.jco.2011.07.001.

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32

El-Basil, S. "Theory and computational applications of Fibonacci graphs." Journal of Mathematical Chemistry 2, no. 1 (1988): 1–29. http://dx.doi.org/10.1007/bf01166466.

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Iskakova, M. T., М. К. Shuakayev, Е. А. Tuiykov, and К. Т. Nazarbekova. "R. KALMAN `S PROBLEM ABOUT FIBONACCI `S NUMBERS." BULLETIN Series of Physics & Mathematical Sciences 72, no. 4 (2020): 28–33. http://dx.doi.org/10.51889/2020-4.1728-7901.04.

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In this paper authors are considered the R. Kalman`s problem about of Fibonacci numbers. An overview of research methods for control theory systems in two concepts “state space” and the “input-output” mapping is presented. In this paper, we consider the problem of R. Kalman on Fibonacci numbers, which consists in the following. R. Kalman's problem on Fibonacci numbers is considered, which is as follows. Fibonacci numbers form a minimal Realization. The authors of the article formulated a theorem, which was given the name of the outstanding American Scientist R. Kalman. The proof of the theorem

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34

Slapničar, Ivan. "On the spectra of generalized Fibonacci and Fibonacci-like operators." Operators and Matrices, no. 1 (2012): 49–62. http://dx.doi.org/10.7153/oam-06-03.

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35

Anisimov, A. V. "Fibonacci hypercube." International Journal of Computer Mathematics 68, no. 3-4 (1998): 221–29. http://dx.doi.org/10.1080/00207169808804690.

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MISHRA, R. K., K. D. MISRA, A. P. MISRA, and A. K. TRIPATHI. "DESIGN AND OPERATING CHARACTERISTICS OF A BANDPASS OPTICAL FILTER WITH OBLIQUE BANDGAP STRUCTURES OF FIBONACCI LATTICES." Surface Review and Letters 12, no. 01 (2005): 47–57. http://dx.doi.org/10.1142/s0218625x05006779.

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In this paper, spectral properties of optical bandpass Fibonacci filter are studied using the transfer matrix approach. The dispersion relation, transmission coefficient of quasiperiodic optical filter structure arranged according to recurrent Fibonacci rule is demonstrated for normal, s-polarized and p-polarized modes of radiation. Numerical calculations for different generations of Fibonacci structure at various angles of incidence are presented. The idea is based on the famous Kroning–Penny model in the band theory of solids. An interesting method for the existence of forbidden photonic obl

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Ddamulira, Mahadi, and Florian Luca. "On the problem of Pillai with k-generalized Fibonacci numbers and powers of 3." International Journal of Number Theory 16, no. 07 (2020): 1643–66. http://dx.doi.org/10.1142/s1793042120500876.

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For an integer [Formula: see text], let [Formula: see text] be the [Formula: see text]-generalized Fibonacci sequence which starts with [Formula: see text] (a total of [Formula: see text] terms) and for which each term afterwards is the sum of the [Formula: see text] preceding terms. In this paper, we find all integers [Formula: see text] with at least two representations as a difference between a [Formula: see text]-generalized Fibonacci number and a power of [Formula: see text]. This paper continues the previous work of the first author for the Fibonacci numbers, and for the Tribonacci numbe

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Prasad, Bandhu. "Coding theory on h(x) extension of m sequences for Fibonacci numbers." Discrete Mathematics, Algorithms and Applications 07, no. 02 (2015): 1550008. http://dx.doi.org/10.1142/s1793830915500081.

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In this paper, we introduced h(x) extension of m sequences of Fibonacci numbers polynomials of order m and a new Sn matrix of order m where h(x)(> 0) is a polynomial with real coefficients. Thereby, we discuss various properties of Sn matrix and the Fibonacci coding theory followed from this matrix. We established the relations among the code matrix elements, error deduction and correction for this coding theory. In general, correct ability potential of this method increases and it is independent of h(x). But h(x) being a polynomial, improves the cryptography protection.

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Bravo, Jhon J., and Jose L. Herrera. "Fibonacci numbers in generalized Pell sequences." Mathematica Slovaca 70, no. 5 (2020): 1057–68. http://dx.doi.org/10.1515/ms-2017-0413.

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AbstractIn this paper, by using lower bounds for linear forms in logarithms of algebraic numbers and the theory of continued fractions, we find all Fibonacci numbers that appear in generalized Pell sequences. Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for Fibonacci numbers in the Pell sequence.

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40

Maclachlan, C., and A. W. Reid. "Generalised Fibonacci manifolds." Transformation Groups 2, no. 2 (1997): 165–82. http://dx.doi.org/10.1007/bf01235939.

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41

Jeffery, Thomas. "Cassini- and Catalan-like formulas for polygonal numbers and simplex numbers." SURG Journal 5, no. 2 (2012): 37–43. http://dx.doi.org/10.21083/surg.v5i2.1725.

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Cassini’s formula and Catalan’s formula are two results from the theory of Fibonacci numbers. This article derives results similar to these, however instead of applying to Fibonacci numbers, they are applied to polygonal numbers and simplex numbers. Triangular numbers are considered first. We then generalize to polygonal and simplex numbers. For polygonal numbers the properties of determinants are used to simplify calculations. For simplex numbers Pascal’s Theorem is used.

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FAN, CHENG-MING, and H. J. SHYR. "SOME PROPERTIES OF FIBONACCI LANGUGAES." Tamkang Journal of Mathematics 27, no. 2 (1996): 165–82. http://dx.doi.org/10.5556/j.tkjm.27.1996.4357.

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   Two particular types of Fibonacci languages $F_{a, b}^1=\{a, b, ab, bab, abbab, \cdots \}$ and $F_{a, b}^0=\{a, b, ba, bab, babba, \cdots \}$    were defined on the free monoid $X^*$ generated by the alphabet $X = \{a, b\}$. In this paper we investigate some algebraic properties of these two types of Fibonacci languages. We show that a general Fibonacci language is a homomorphical image of either $F_{a, b}^1$ or $F_{a, b}^0$. We also study the properties of Fibonacci language related to formal language theory and codes We obtained the facts that ev

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Basu, Manjusri, and Monojit Das. "Coding theory on generalized Fibonacci n-step polynomials." Journal of Information and Optimization Sciences 38, no. 1 (2017): 83–131. http://dx.doi.org/10.1080/02522667.2016.1160618.

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Wang, Chumin, and R. A. Barrio. "Theory of the Raman Response in Fibonacci Superlattices." Physical Review Letters 61, no. 2 (1988): 191–94. http://dx.doi.org/10.1103/physrevlett.61.191.

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Falcón, Sergio, and Ángel Plaza. "Iterated Partial Sums of the k-Fibonacci Sequences." Axioms 11, no. 10 (2022): 542. http://dx.doi.org/10.3390/axioms11100542.

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In this paper, we find the sequence of partial sums of the k-Fibonacci sequence, say, Sk,n=∑j=1nFk,j, and then we find the sequence of partial sums of this new sequence, Sk,n2)=∑j=1nSk,j, and so on. The iterated partial sums of k-Fibonacci numbers are given as a function of k-Fibonacci numbers, in powers of k, and in a recursive way. We finish the topic by indicating a formula to find the first terms of these sequences from the k-Fibonacci numbers themselves.

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BASU, MANJUSRI, and MONOJIT DAS. "TRIBONACCI MATRICES AND A NEW CODING THEORY." Discrete Mathematics, Algorithms and Applications 06, no. 01 (2014): 1450008. http://dx.doi.org/10.1142/s1793830914500086.

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In this paper, we consider the series of Tribonacci numbers. Thereby, we introduce a new coding theory called Tribonacci coding theory based on Tribonacci numbers and show that in the simplest case, the correct ability of this method is 99.80% whereas the correct ability of the Fibonacci coding/decoding method is 93.33%.

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Kumar, Nand Kishor, and Kripa Sindhu Prasad. "Exploring Fibonacci Algebraic Equations and Their Application." Cognition 7, no. 1 (2025): 45–53. https://doi.org/10.3126/cognition.v7i1.74741.

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This article explores the sequence's generating functions, algebraic equations regulating it, and relationships to abstract algebra. Fibonacci algebra shows how patterns, numbers, and equations interact, from quadratic equations and generating functions to matrix algebra and beyond. Its simplicity and the surprising breadth of its linkages to other areas of mathematics are what give it its ongoing appeal. The study of Fibonacci algebraic equations is not just an exploration of an ancient sequence but a gateway to profound mathematical structures and applications. Whether through its connection

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Sivaraman, Dr R. "Generating Matrices and Fibonacci-Like Numbers." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 2 (2021): 2901–4. http://dx.doi.org/10.17762/turcomat.v12i2.2325.

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Among several classes of interesting numbers, Fibonacci numbers plays a significant role and these numbers very often turns up in several branches of Science, Engineering and Technology. Considering two by two square matrices whose entries are Fibonacci and Fibonacci like sequence numbers, we investigate the integral powers of such matrices and prove interesting results concerning them. Finally, the limiting cases of entries of nth power of such matrices are derived.

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WILLIAMS, GERALD. "GROUPS OF FIBONACCI TYPE REVISITED." International Journal of Algebra and Computation 22, no. 08 (2012): 1240002. http://dx.doi.org/10.1142/s0218196712400024.

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This article concerns a class of groups of Fibonacci type introduced by Johnson and Mawdesley that includes Conway's Fibonacci groups, the Sieradski groups, and the Gilbert–Howie groups. This class of groups provides an interesting focus for developing the theory of cyclically presented groups and, following questions by Bardakov and Vesnin and by Cavicchioli, Hegenbarth, and Repovš, they have enjoyed renewed interest in recent years. We survey results concerning their algebraic properties, such as isomorphisms within the class, the classification of the finite groups, small cancellation prope

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SANNA, CARLO. "THE QUOTIENT SET OF -GENERALISED FIBONACCI NUMBERS IS DENSE IN." Bulletin of the Australian Mathematical Society 96, no. 1 (2017): 24–29. http://dx.doi.org/10.1017/s0004972716001118.

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The quotient set of $A\subseteq \mathbb{N}$ is defined as $R(A):=\{a/b:a,b\in A,b\neq 0\}$. Using algebraic number theory in $\mathbb{Q}(\sqrt{5})$, Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly, to appear] proved that the quotient set of Fibonacci numbers is dense in the $p$-adic numbers $\mathbb{Q}_{p}$ for all prime numbers $p$. For any integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq -(k-2)}$ be the sequence of $k$-generalised Fibonacci numbers, defined by the initial values $0,0,\ldots ,0,1$ ($k$ terms) and such that each successive term is the sum of the $k$ precedin

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