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Journal articles on the topic 'Fibonacci numbers'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

KÖKEN, Fikri, and Emre KANKAL. "Altered Numbers of Fibonacci Number Squared." Journal of New Theory, no. 45 (December 31, 2023): 73–82. http://dx.doi.org/10.53570/jnt.1368751.

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We investigate two types of altered Fibonacci numbers obtained by adding or subtracting a specific value $\{a\}$ from the square of the $n^{th}$ Fibonacci numbers $G^{(2)}_{F(n)}(a)$ and $H^{(2)}_{F(n)}(a)$. These numbers are significant as they are related to the consecutive products of the Fibonacci numbers. As a result, we establish consecutive sum-subtraction relations of altered Fibonacci numbers and their Binet-like formulas. Moreover, we explore greatest common divisor (GCD) sequences of r-successive terms of altered Fibonacci numbers represented by $\left\{G^{(2)}_{F(n), r}(a)\right\}$

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2

Shannon, Anthony G., Özgür Erdağ, and Ömür Deveci. "On the connections between Pell numbers and Fibonacci p-numbers." Notes on Number Theory and Discrete Mathematics 27, no. 1 (2021): 148–60. http://dx.doi.org/10.7546/nntdm.2021.27.1.148-160.

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In this paper, we define the Fibonacci–Pell p-sequence and then we discuss the connection of the Fibonacci–Pell p-sequence with the Pell and Fibonacci p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Fibonacci–Pell p-numbers by the aid of the n-th power of the generating matrix of the Fibonacci–Pell p-sequence. Furthermore, we derive relationships between the Fibonacci–Pell p-numbers and their permanent, determinant and sums of certain matrices.

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3

Howard, Fredric T. "Fibonacci numbers." Mathematical Intelligencer 26, no. 1 (2004): 65. http://dx.doi.org/10.1007/bf02985406.

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4

Kortezov, Ivaylo. "Double Counting and Fibonacci Numbers." Mathematics and Informatics 67, no. 3 (2024): 336–50. https://doi.org/10.53656/math2024-3-7-dou.

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The article proposes a double counting method for proving combinatorial identities involving Fibonacci numbers. The method is easily remembered by the students. It allows them to apply it creatively in many situations, by (re)discovering the results by themselves and asking meaningful questions. The method is mainly oriented towards preparation for mathematical competitions.

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5

Wituła, Roman, та Damian Słota. "δ-Fibonacci numbers". Applicable Analysis and Discrete Mathematics 3, № 2 (2009): 310–29. http://dx.doi.org/10.2298/aadm0902310w.

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The scope of the paper is the definition and discussion of the polynomial generalizations of the Fibonacci numbers called here ?-Fibonacci numbers. Many special identities and interesting relations for these new numbers are presented. Also, different connections between ?-Fibonacci numbers and Fibonacci and Lucas numbers are proven in this paper.

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6

Tasyurdu, Yasemin. "Generalized Fibonacci numbers with five parameters." Thermal Science 26, Spec. issue 2 (2022): 495–505. http://dx.doi.org/10.2298/tsci22s2495t.

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In this paper, we define five parameters generalization of Fibonacci numbers that generalizes Fibonacci, Pell, Modified Pell, Jacobsthal, Narayana, Padovan, k-Fibonacci, k-Pell, Modified k-Pell, k-Jacobsthal numbers and Fibonacci p-numbers, distance Fibonacci numbers, (2, k)-distance Fibonacci numbers, generalized (k, r)-Fibonacci numbers in the distance sense by extending the definition of a distance in the recurrence relation with two parameters and adding three parameters in the definition of this distance, simultaneously. Tiling and combinatorial interpretations of generalized Fibonacci nu

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7

Szynal-Liana, Anetta, Andrzej Włoch, and Iwona Włoch. "On Types of Distance Fibonacci Numbers Generated by Number Decompositions." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/491591.

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We introduce new types of distance Fibonacci numbers which are closely related with number decompositions. Using special decompositions of the numbernwe give a sequence of identities for them. Moreover, we give matrix generators for distance Fibonacci numbers and their direct formulas.

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8

Bednarz, Natalia. "On (k,p)-Fibonacci Numbers." Mathematics 9, no. 7 (2021): 727. http://dx.doi.org/10.3390/math9070727.

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In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultaneously. We prove some identities which generalize well-known relations for Fibonacci numbers, Pell numbers and their generalizations. A matrix representation for generalized Fibonacci numbers is given, too.

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9

Han, Jeong, Hee Kim, and Joseph Neggers. "On Fibonacci functions with Fibonacci numbers." Advances in Difference Equations 2012, no. 1 (2012): 126. http://dx.doi.org/10.1186/1687-1847-2012-126.

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10

Luca, Florian, and V. Janitzio Mejía Huguet. "Fibonacci-Riesel and Fibonacci-Sierpiński numbers." Fibonacci Quarterly 46-47, no. 3 (2008): 198–201. http://dx.doi.org/10.1080/00150517.2008.12428153.

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11

Laugier, Alexandre, and Manjil P. Saikia. "Some Properties of Fibonacci Numbers, Generalized Fibonacci Numbers and Generalized Fibonacci Polynomial Sequences." Kyungpook mathematical journal 57, no. 1 (2017): 1–84. http://dx.doi.org/10.5666/kmj.2017.57.1.1.

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12

Herrera, Jose L., Jhon J. Bravo, and Carlos A. Gómez. "Curious Generalized Fibonacci Numbers." Mathematics 9, no. 20 (2021): 2588. http://dx.doi.org/10.3390/math9202588.

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A generalization of the well-known Fibonacci sequence is the k−Fibonacci sequence whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i.e., numbers whose base ten representation have the form a⋯ab⋯ba⋯a). This work continues and extends the prior result of Trojovský, who found all Fibonacci numbers with a prescribed block of digits, and the result of Alahmadi et al., who searched for k-Fibonacci numbers, which are concatenation of two repdigits.

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13

KARATAŞ, Adnan. "Shifted Fibonacci Numbers." Afyon Kocatepe University Journal of Sciences and Engineering 23, no. 6 (2023): 1440–44. http://dx.doi.org/10.35414/akufemubid.1345862.

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Shifted Fibonacci numbers have been examined in the literature in terms of the greatest common divisor, but appropriate definitions and fundamental equations have not been worked on. In this article, we have obtained the Binet formula, which is a fundamental equation used to obtain the necessary element of the shifted Fibonacci number sequence. Additionally, we have obtained many well-known identities such as Cassini, Honsberger, and various other identities for this sequence. Furthermore, summation formulas for shifted Fibonacci numbers have been presented.

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14

Adhikari, Iswar Mani. "On the Fibonacci and the Generalized Fibonacci Sequence." Journal of Nepal Mathematical Society 8, no. 1 (2025): 30–38. https://doi.org/10.3126/jnms.v8i1.80311.

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Fibonacci numbers and their sequence are found abundantly in nature. There is a close relation among the Golden, Fibonacci, and Lucas ratios. Such ratios are inherent to design, architecture, construction, and even to the beauty of different natural and manmade solid objects. Other variants, like the k-generalized Fibonacci sequence, the Fibonacci p-numbers, Lucas numbers, and the ordinary Fibonacci numbers, have some interesting properties and are based on recurrent relations. This article describes the different structures, mathematical beauty, and identities related to such sequences and nu

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15

Ramírez, José L. "Some Properties of Convolved k-Fibonacci Numbers." ISRN Combinatorics 2013 (June 12, 2013): 1–5. http://dx.doi.org/10.1155/2013/759641.

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We define the convolved k-Fibonacci numbers as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the k-Fibonacci and k-Lucas numbers. Moreover we obtain the convolved k-Fibonacci numbers from a family of Hessenberg matrices.

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16

Gültekin, Inci, and Ömür Deveci. "On the arrowhead-Fibonacci numbers." Open Mathematics 14, no. 1 (2016): 1104–13. http://dx.doi.org/10.1515/math-2016-0100.

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AbstractIn this paper, we define the arrowhead-Fibonacci numbers by using the arrowhead matrix of the characteristic polynomial of the k-step Fibonacci sequence and then we give some of their properties. Also, we study the arrowhead-Fibonacci sequence modulo m and we obtain the cyclic groups from the generating matrix of the arrowhead-Fibonacci numbers when read modulo m. Then we derive the relationships between the orders of the cyclic groups obtained and the periods of the arrowhead-Fibonacci sequence modulo m.

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17

Tasci, Dursun, Mirac Cetin Firengiz, and Naim Tuglu. "Incomplete Bivariate Fibonacci and Lucas -Polynomials." Discrete Dynamics in Nature and Society 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/840345.

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We define the incomplete bivariate Fibonacci and Lucas polynomials. In the case , , we obtain the incomplete Fibonacci and Lucas numbers. If , , we have the incomplete Pell and Pell-Lucas numbers. On choosing , , we get the incomplete generalized Jacobsthal number and besides for the incomplete generalized Jacobsthal-Lucas numbers. In the case , , , we have the incomplete Fibonacci and Lucas numbers. If , , , , we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas polynomials are given.

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18

Liptai, Kálmán. "Fibonacci Balancing Numbers." Fibonacci Quarterly 42, no. 4 (2004): 330–40. http://dx.doi.org/10.1080/00150517.2004.12428404.

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19

McDaniel, Wayne L. "Pronic Fibonacci Numbers." Fibonacci Quarterly 36, no. 1 (1998): 56–59. http://dx.doi.org/10.1080/00150517.1998.12428961.

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20

Safran, Charles. "The Fibonacci Numbers." CHANCE 5, no. 1-2 (1992): 43–46. http://dx.doi.org/10.1080/09332480.1992.11882462.

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21

Gómez Ruiz, Carlos Alexis, and Florian Luca. "Fibonacci factoriangular numbers." Indagationes Mathematicae 28, no. 4 (2017): 796–804. http://dx.doi.org/10.1016/j.indag.2017.05.002.

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22

Altassan, Alaa, and Murat Alan. "Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers." Mathematica Slovaca 74, no. 3 (2024): 563–76. http://dx.doi.org/10.1515/ms-2024-0042.

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Abstract Let (Fn ) n≥0 and (Ln ) n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn = 10 d Fm + Lk and Fn = 10 d Lm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk , respectively. We use

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23

Jugal, Kishore, and Verma Vipin. "Some Identities on Sums of Finite Products of the Pell, Fibonacci, and Chebyshev Polynomials." Indian Journal of Science and Technology 16, no. 12 (2023): 941–55. https://doi.org/10.17485/IJST/v16i12.2213.

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Abstract <strong>Objectives:</strong> This study will introduce some new identities for sums of finite products of the Pell, Fibonacci, and Chebyshev polynomials in terms of derivatives of Pell polynomials. Similar identities for Fibonacci and Lucas numbers will be deduced.<strong> Methods:</strong> Results are obtained by using differential calculus, combinatory computations, and elementary algebraic computations. <strong>Findings:</strong> In terms of derivatives of Pell polynomials, identities on sums of finite products of the Fibonacci numbers, Lucas numbers, Pell

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24

Singh, Bijendra, Kiran Sisodiya, and Farooq Ahmad. "On the Products ofk-Fibonacci Numbers andk-Lucas Numbers." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/505798.

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In this paper we investigate some products ofk-Fibonacci andk-Lucas numbers. We also present some generalized identities on the products ofk-Fibonacci andk-Lucas numbers to establish connection formulas between them with the help of Binet's formula.

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25

Shannon, Anthony G., Irina Klamka, and Robert van Gend. "Generalized Fibonacci Numbers and Music." JOURNAL OF ADVANCES IN MATHEMATICS 14, no. 1 (2018): 7564–79. http://dx.doi.org/10.24297/jam.v14i1.7323.

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Mathematics and music have well documented historical connections. Just as the ordinary Fibonacci numbers have links with the golden ratio, this paper considers generalized Fibonacci numbers developed from generalizations of the golden ratio. It is well known that the Fibonacci sequence of numbers underlie certain musical intervals and compositions but to what extent are these connections accidental or structural, coincidental or natural and do generalized Fibonacci numbers share any of these connections?

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26

Özkan, Engin, and Bahar Kuloğlu. "On the bicomplex Gaussian Fibonacci and Gaussian Lucas numbers." Acta et Commentationes Universitatis Tartuensis de Mathematica 26, no. 1 (2022): 33–43. http://dx.doi.org/10.12697/acutm.2022.26.03.

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We give the bicomplex Gaussian Fibonacci and the bicomplex Gaussian Lucas numbers and establish the generating functions and Binet’s formulas related to these numbers. Also, we present the summation formula, matrix representation and Honsberger identity and their relationship between these numbers. Finally, we show the relationships among the bicomplex Gaussian Fibonacci, the bicomplex Gaussian Lucas, Gaussian Fibonacci, Gaussian Lucas and Fibonacci numbers.

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27

Bród, Dorota. "On Distance(r,k)-Fibonacci Numbers and Their Combinatorial and Graph Interpretations." Journal of Applied Mathematics 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/879510.

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We introduce three new two-parameter generalizations of Fibonacci numbers. These generalizations are closely related tok-distance Fibonacci numbers introduced recently. We give combinatorial and graph interpretations of distance(r,k)-Fibonacci numbers. We also study some properties of these numbers.

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28

Boughaba, Souhila, Ali Boussayoud, Serkan Araci, Mohamed Kerada, and Mehmet Acikgoz. "Construction of a new class of generating functions of binary products of some special numbers and polynomials." Filomat 35, no. 3 (2021): 1001–13. http://dx.doi.org/10.2298/fil2103001b.

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In this paper, we derive some new symmetric properties of k-Fibonacci numbers by making use of symmetrizing operator. We also give some new generating functions for the products of some special numbers such as k-Fibonacci numbers, k-Pell numbers, Jacobsthal numbers, Fibonacci polynomials and Chebyshev polynomials.

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29

Tas, Sait. "The hyperbolic-type k-Fibonacci sequences and their applications." Thermal Science 26, Spec. issue 2 (2022): 551–58. http://dx.doi.org/10.2298/tsci22s2551t.

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In this study, we define hyperbolic-type k-Fibonacci numbers and then give the relationships between the k-step Fibonacci numbers and the hyperbolic-type k-Fibonacci numbers. In addition, we study the hyperbolic-type k-Fibonacci sequence modulo m and then we give periods of the Hperbolic-type k-Fibonacci sequences for any k and m which are related the periods of the k-step Fibonacci sequences modulo m. Furthermore, we extend the hyperbolic-type k-Fibonacci sequences to groups. Finally, we obtain the periods of the hyperbolic-type 2-Fibonacci sequences in the dihedral group D2m, (m ? 2) with re

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30

Adegoke, Kunle. "Fibonacci series from power series." Notes on Number Theory and Discrete Mathematics 27, no. 3 (2021): 44–62. http://dx.doi.org/10.7546/nntdm.2021.27.3.44-62.

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We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

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31

Yılmaz, Nazmiye, Ali Aydoğdu, and Engin Özkan. "Some properties of k-generalized Fibonacci numbers." Mathematica Montisnigri 50 (2021): 73–79. http://dx.doi.org/10.20948/mathmontis-2021-50-7.

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In the present paper, we propose some properties of the new family 𝑘-generalized Fibonacci numbers which related to generalized Fibonacci numbers. Moreover, we give some identities involving binomial coefficients for 𝑘-generalized Fibonacci numbers.

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32

Raman, Indhumathi. "A Note on Closed-Form Representation of Fibonacci Numbers Using Fibonacci Trees." ISRN Discrete Mathematics 2014 (February 12, 2014): 1–3. http://dx.doi.org/10.1155/2014/132925.

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We give a new representation of the Fibonacci numbers. This is achieved using Fibonacci trees. With the help of this representation, the nth Fibonacci number can be calculated without having any knowledge about the previous Fibonacci numbers.

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33

Lewis, Barry. "Trigonometry and Fibonacci numbers." Mathematical Gazette 91, no. 521 (2007): 216–26. http://dx.doi.org/10.1017/s0025557200181550.

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This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

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34

BELAGGOUN, NASSIMA, and HACÈNE BELBACHIR. "Bi-Periodic Hyper-Fibonacci Numbers." Kragujevac Journal of Mathematics 49, no. 4 (2024): 603–14. http://dx.doi.org/10.46793/kgjmat2504.603b.

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In the present paper, we introduce and study a new generalization of hyper-Fibonacci numbers, called the bi-periodic hyper-Fibonacci numbers. Furthermore, we give a combinatorial interpretation using the weighted tilings approach and prove several identities relating these numbers. Moreover, we derive their generating function and new identities for the classical hyper-Fibonacci numbers.

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35

Akbiyik, Mücahit, and Jeta Alo. "On Third-Order Bronze Fibonacci Numbers." Mathematics 9, no. 20 (2021): 2606. http://dx.doi.org/10.3390/math9202606.

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In this study, we firstly obtain De Moivre-type identities for the second-order Bronze Fibonacci sequences. Next, we construct and define the third-order Bronze Fibonacci, third-order Bronze Lucas and modified third-order Bronze Fibonacci sequences. Then, we define the generalized third-order Bronze Fibonacci sequence and calculate the De Moivre-type identities for these sequences. Moreover, we find the generating functions, Binet’s formulas, Cassini’s identities and matrix representations of these sequences and examine some interesting identities related to the third-order Bronze Fibonacci se

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36

Senad, Orhani. "Fibonacci Numbers as a Natural Phenomenon." International Journal of Scientific Research and Innovative Studies 1, no. 1 (2022): 7–13. https://doi.org/10.5281/zenodo.6869475.

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This paper presents an attempt to explain and experiment with Fibonacci numbers. It is illustrated with examples and concrete cases many of the properties of these numbers which are often found in practice or even are things that we see and touch every day. My goal has been to make a summary of writings, analyzes, studies, games, natural events as well as research by means of examples for famous Fibonacci numbers. The Fibonacci sequence can be observed in a variety of stunning phenomena in nature. Therefore, this study first presents the Fibonacci sequence and also describes some of the Fibona

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37

Deveci, Ömür. "On the connections among Fibonacci, Pell, Jacobsthal and Padovan numbers." Notes on Number Theory and Discrete Mathematics 27, no. 2 (2021): 111–28. http://dx.doi.org/10.7546/nntdm.2021.27.2.111-128.

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In this paper, we define the Fibonacci–Jacobsthal, Padovan–Fibonacci, Pell–Fibonacci, Pell–Jacobsthal, Padovan–Pell and Padovan–Jacobsthal sequences which are directly related with the Fibonacci, Jacobsthal, Pell and Padovan numbers and give their structural properties by matrix methods. Then we obtain new relationships between Fibonacci, Jacobsthal, Pell and Padovan numbers.

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38

Pethe, S., and A. F. Horadam. "Generalised Gaussian Fibonacci numbers." Bulletin of the Australian Mathematical Society 33, no. 1 (1986): 37–48. http://dx.doi.org/10.1017/s0004972700002847.

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In this paper, generalised Gaussian Fibonacci numbers are defined and, using the recurrence relation satisfied by them, we obtain a number of summation identities involving the products of combinations of Fibonacci, Pell and Chebyshev polynomials.

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39

Martinsen, Thor. "Non-Fisherian generalized Fibonacci numbers." Notes on Number Theory and Discrete Mathematics 31, no. 2 (2025): 370–89. https://doi.org/10.7546/nntdm.2025.31.2.370-389.

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Using biology as inspiration, this paper explores a generalization of the Fibonacci sequence that involves gender biased sexual reproduction. The female, male, and total population numbers along with their associated recurrence relations are considered. We demonstrate that the generalized Fibonacci numbers being investigated are generalized third order Pell–Lucas numbers. Sequence properties, generating functions, and closed-form solutions for these new generalized Fibonacci numbers, as well as several identities involving Jacobsthal, Leonardo, and generalized Leonardo numbers are presented. T

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40

Yilmaz, Nazmiye. "More identities on Fibonacci and Lucas hybrid numbers." Notes on Number Theory and Discrete Mathematics 27, no. 2 (2021): 159–67. http://dx.doi.org/10.7546/nntdm.2021.27.2.159-167.

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We give several identities about Fibonacci and Lucas hybrid numbers. We introduce the Fibonacci and Lucas hybrid numbers with negative subscripts. We obtain different Cassini identities for the conjugate of the Fibonacci and Lucas hybrid numbers by two different determinant definitions of a hybrid square matrix (whose entries are hybrid numbers).

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41

Battaloglu, Rifat, and Yilmaz Simsek. "On New Formulas of Fibonacci and Lucas Numbers Involving Golden Ratio Associated with Atomic Structure in Chemistry." Symmetry 13, no. 8 (2021): 1334. http://dx.doi.org/10.3390/sym13081334.

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The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many new relations among the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers. Moreover, some applications of the Fibonacci numbers and the golden ratio in chemistry are given.

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42

GÖKBAŞ, Hasan. "Gaussian Quaternions Including Biperiodic Fibonacci and Lucas Numbers." Osmaniye Korkut Ata Üniversitesi Fen Bilimleri Enstitüsü Dergisi 6, no. 1 (2023): 594–604. http://dx.doi.org/10.47495/okufbed.1117644.

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In this study, we define a type of bi-periodic Fibonacci and Lucas numbers which are called bi-periodic Fibonacci and Lucas Gaussian quaternions. We also give the relationship between negabi-periodic Fibonacci and Lucas Gaussian quaternions and bi-periodic Fibonacci and Lucas Gaussian quaternions. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity, like-Tagiuri’s identity, Honberger’s identity and some formulas for these new type numbers. Some algebraic proporties of bi-periodic Fibonacci and Lucas Gaussian quaternions whic

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43

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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44

Shannon, Anthony G., and Seamus A. Power. "Natural Mathematics, the Fibonacci Numbers and Aesthetics in Art." JOURNAL OF ADVANCES IN MATHEMATICS 17 (October 28, 2019): 248–54. http://dx.doi.org/10.24297/jam.v17i0.8479.

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The Mathematics of beauty and beauty in mathematics are important ingredients in learning in the liberal arts. The Fibonacci numbers play an important and useful role in this. This paper seeks to present and illustrate a grounding of visual aesthetics in natural mathematical principles, centered upon the Fibonacci numbers. The specific natural mathematical principles investigated are the Fibonacci numbers, the Fibonacci Spiral, and the Cosmic Bud.

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45

Veenstra, Tamara B., and Catherine M. Miller. "The Matrix Connection: Fibonacci and Inductive Proof." Mathematics Teacher 99, no. 5 (2006): 328–33. http://dx.doi.org/10.5951/mt.99.5.0328.

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This article presents several activities (some involving graphing calculators) designed to guide students to discover several interesting properties of Fibonacci numbers. Then, we explore interesting connections between Fibonacci numbers and matrices; using this connection and induction we prove divisibility properties of Fibonacci numbers. Includes problems and samples of tasks used to help student discover patterns within the Fibonacci Sequence and connections to matrix algebra.

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46

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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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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Cihan, Arzu, Ayşe Zeynep Azak, Mehmet Ali Güngör, and Murat Tosun. "A Study on Dual Hyperbolic Fibonacci and Lucas Numbers." Analele Universitatii "Ovidius" Constanta - Seria Matematica 27, no. 1 (2019): 35–48. http://dx.doi.org/10.2478/auom-2019-0002.

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Abstract In this study, the dual-hyperbolic Fibonacci and dual-hyperbolic Lucas numbers are introduced. Then, the fundamental identities are proven for these numbers. Additionally, we give the identities regarding negadual-hyperbolic Fibonacci and negadual-hyperbolic Lucas numbers. Finally, Binet formulas, D’Ocagne, Catalan and Cassini identities are obtained for dual-hyperbolic Fibonacci and dual-hyperbolic Lucas numbers.

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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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Kosoboutskyy, Petro, Mariana Karkulovska, and Yuliia Losynska. "REGULARITIES OF NUMBERS IN THE FIBONACHI TRIANGLE CONSTRUCTED ON THE DEGREE TRANSFORMATIONS OF A SQUARE THREE MEMBERS." Computer Design Systems. Theory and Practice 3, no. 1 (2021): 37–44. http://dx.doi.org/10.23939/cds2021.01.037.

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In this paper, it is shown that the Fibonacci triangle is formed from the elements of power transformations of a quadratic trinomial. It is binary structured by domains of rows of equal lengths, in which the sum of numbers forms a sequence of certain numbers. This sequence coincides with the transformed bisection of the classical sequence of Fibonacci numbers. The paper substantiates Pascal's rule for calculating elements in the lines of a Fibonacci triangle. The general relations of two forgings of numbers in lines of a triangle of Fibonacci for arbitrary values are received

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