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Journal articles on the topic 'Fibonacci tiling'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Lifshitz, Ron. "The square Fibonacci tiling." Journal of Alloys and Compounds 342, no. 1-2 (2002): 186–90. http://dx.doi.org/10.1016/s0925-8388(02)00169-x.

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2

TAŞYURDU, Yasemin, and Berke CENGİZ. "A TILING APPROACH TO FIBONACCI p-NUMBERS." Journal of Universal Mathematics 5, no. 2 (2022): 177–84. http://dx.doi.org/10.33773/jum.1142766.

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In this paper, we introduce tiling representations of Fibonacci p-numbers, which are generalizations of the well-known Fibonacci and Narayana numbers, and generalized in the distance sense. We obtain Fibonacci p-numbers count the number of distinct ways to tile a 1 × n board using various 1 × r, r-ominoes from r = 1 up to r = p + 1. Moreover, the product identities and sum formulas of these numbers with special subscripts are given by tiling interpretations that allow the derivation of their properties.

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3

Margenstern, Maurice. "Fibonacci Type Coding for the Regular Rectangular Tilings of the Hyperbolic Plane." JUCS - Journal of Universal Computer Science 9, no. (5) (2003): 398–422. https://doi.org/10.3217/jucs-009-05-0398.

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The study of cellular automata (CA) on tilings of hyperbolic plane was initiated in [6]. Appropriate tools were developed which allow us to produce linear algorithms to implement cellular automata on the tiling of the hyperbolic plane with the regular rectangular pentagons, [8, 10]. In this paper we modify and improve these tools, generalise the algorithms and develop them for tilings of the hyperbolic plane with regular rectangular s-gons for s 5. For this purpose a combinatorial structure of these tilings is studied.

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4

Huegy, Charles W., and Douglas B. West. "A Fibonacci tiling of the plane." Discrete Mathematics 249, no. 1-3 (2002): 111–16. http://dx.doi.org/10.1016/s0012-365x(01)00239-4.

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5

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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6

Amaral, Marcelo, David Chester, Fang Fang, and Klee Irwin. "Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing." Symmetry 14, no. 9 (2022): 1780. http://dx.doi.org/10.3390/sym14091780.

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The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing. A topological quantum computing platform promises to deliver more robust qubits with additional hardware-level protection against errors that could lead to the desired large-scale quantum computation. We propose quasicrystal materials as such a natural platform and show that they exhibit anyonic behavior that can be used for topological quantum computing. Different from anyons, quasicrystals are already implemented in lab

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7

Artz, Jacob, and Michael Rowell. "A tiling approach to Fibonacci product identities." Involve, a Journal of Mathematics 2, no. 5 (2010): 581–87. http://dx.doi.org/10.2140/involve.2009.2.581.

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8

Gähler, F., and E. Miro. "Topology of the Random Fibonacci Tiling Space." Acta Physica Polonica A 126, no. 2 (2014): 564–67. http://dx.doi.org/10.12693/aphyspola.126.564.

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9

BERKOFF, A. M., J. M. HENLE, A. E. MCDONOUGH, and A. P. WESOLOWSKI. "POSSIBILITIES AND IMPOSSIBILITIES IN SQUARE-TILING." International Journal of Computational Geometry & Applications 21, no. 05 (2011): 545–58. http://dx.doi.org/10.1142/s0218195911003792.

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A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of sidelength n for every n in the set. From Ref. 8 we know that ℕ itself tiles the plane. From that and Ref. 9 we know that the set of even numbers tiles the plane while the set of odd numbers doesn't. In this paper we explore the nature of this property. We show, for example, that neither tiling nor non-tiling is preserved by superset. We show that a set with one or three odd numbers may tile the plane—but a set with two odd numbers can't. We find examples of both tiling and non-tiling se

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10

BEN-ABRAHAM, S. I. "DEFECTIVE VERTEX CONFIGURATIONS IN QUASICRYSTALLINE STRUCTURES." International Journal of Modern Physics B 07, no. 06n07 (1993): 1415–25. http://dx.doi.org/10.1142/s0217979293002407.

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Defective vertex configurations are important for the whole range of models for quasicrystalline structures from quasiperiodic tilings through random tilings to polyhedral glasses. The combinatorially possible vertex configurations are enumerated for the 1D Fibonacci chain, for the 2D Penrose pattern with its generalizations, as well as for the Beenker pattern and the triangle pattern, and for the 3D simple icosahedral tiling. The methods for quantifying the deviation of vertex configurations from perfection are reviewed. The simple method of partial dual overlap provides a means to estimate t

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11

ILAN, RONI, EDO LIBERTY, SHAHAR EVEN-DAR MANDEL, and RON LIFSHITZ. "Electrons and Phonons on the Square Fibonacci Tiling." Ferroelectrics 305, no. 1 (2004): 15–19. http://dx.doi.org/10.1080/00150190490462252.

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12

Urban, Grzegorz, and Janusz Wolny. "Average unit cell of a square Fibonacci tiling." Journal of Non-Crystalline Solids 334-335 (March 2004): 105–9. http://dx.doi.org/10.1016/j.jnoncrysol.2003.11.022.

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13

Fang, Hanzhang. "Three-dimensional tilings of blocks and bracelets and related {2,1,2} sequences." Applied and Computational Engineering 19, no. 1 (2023): 90–102. http://dx.doi.org/10.54254/2755-2721/19/20231014.

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This paper discusses the combinatorial interpretation of the H_n numbers, where each Hn denotes the number of ways to tile a 2 2 n block with 2 2 1 plates and 6-block L shapes. It then investigates a closely related tiling sequence, which is tiling a 22n bracelet with the same two building blocks, and discusses its relation with Hn. The recursive equation for both integer sequences are found using one to one correspondence, induction and Newtons Sum. Additionally, in the case of H_n numbers, its related Lucas Style sequence P_n is found. The relationship between the P_n numbers, the Bn numbers

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14

Alves, Francisco Régis Vieira, and Renata Teófilo de Sousa. "Some Elementary Combinatory Properties and Fibonacci Numbers." Journal of Instructional Mathematics 4, no. 1 (2023): 52–65. http://dx.doi.org/10.37640/jim.v4i1.1756.

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In general, in the midst of History of Mathematics textbooks, we are faced with a discussion due to curiosity about the emblematic Fibonacci Sequence, whose popularization occurred with the proposition of the reproduction model of immortal rabbits. On the other hand, in the comparison of the multiple approaches and discussions of certain subjects in Elementary Mathematics, in the present work, we highlight combinatorial interpretations that, with the support of a characteristic and fundamental reasoning for the mathematics teacher, can be generalized and formalize some eminently intuitive comp

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15

Even-Dar Mandel, S., and R. Lifshitz. "Electronic energy spectra and wave functions on the square Fibonacci tiling." Philosophical Magazine 86, no. 6-8 (2006): 759–64. http://dx.doi.org/10.1080/14786430500313846.

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16

Baake, Michael, and Uwe Grimm. "Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction." Acta Crystallographica Section A Foundations and Advances 76, no. 5 (2020): 559–70. http://dx.doi.org/10.1107/s2053273320007421.

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Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the

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17

Al-Siyabi, Abeer, Nazife Ozdes Koca, and Mehmet Koca. "Icosahedral Polyhedra from D6 Lattice and Danzer’s ABCK Tiling." Symmetry 12, no. 12 (2020): 1983. http://dx.doi.org/10.3390/sym12121983.

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It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3, its roots, and weights are determined in terms of those of D6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H3, and it is shown that the inflation of the tiles can be obtained as p

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18

Wolny, Janusz, Ireneusz Buganski, Maciej Chodyn, Bartlomiej Kozakowski, Pawel Kuczera, and Radoslaw Strzalka. "The phase problem for quasicrystals in reciprocal space." Acta Crystallographica Section A Foundations and Advances 70, a1 (2014): C1197. http://dx.doi.org/10.1107/s2053273314088020.

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The phase problem is very well-known in crystallography and is particularly important for structure solution of quasicrystals. Structure solution ( initial phasing of the diffraction pattern) is the first step of atomic structure determination against the diffraction data. Many tools for solving the phase problem in crystallography were developed over the years. Besides the pioneer Patherson function method or direct methods, also the low density elimination method and, more recently, maximum entropy method or charge flipping algorithm are widely used. We propose another way of phase recovery,

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19

Franco, B. J. O. "Third-order Fibonacci sequence associated to a heptagonal quasiperiodic tiling of the plane." Physics Letters A 178, no. 1-2 (1993): 119–22. http://dx.doi.org/10.1016/0375-9601(93)90737-k.

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20

Franco, B. J. O., J. R. Faleiro Ferreira, and F. W. O. da Silva. "A Third-Order Fibonacci Sequence Associated to a Heptagonal Quasiperiodic Tiling of the Plane." physica status solidi (b) 182, no. 2 (1994): K57—K62. http://dx.doi.org/10.1002/pssb.2221820232.

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21

Strzałka, Radosław, Łukasz Chuchra, and Janusz Wolny. "Envelope Function Analysis of Quasicrystals." Crystals 12, no. 4 (2022): 536. http://dx.doi.org/10.3390/cryst12040536.

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Quasicrystals have attracted a growing interest in material science because of their unique properties and applications. Proper determination of the atomic structure is important in designing a useful application of these materials, for which a difficult phase problem of the structure factor must be solved. Diffraction patterns of quasicrystals consist of a periodic series of peaks, which can be reduced to a single envelope. Knowing the distribution of the diffraction image into series, it is possible to recover information about the phase of the structure factor without using time-consuming i

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22

Jalal, Khairabadi Bahram Sadeghi Bigham Rebvar Hosseini Zohreh Mohammad Alizadeh. "Tiling a Rectangular Area Using a Set of Unique Squares." International Journal of Computer & Information Technologies (IJOCIT) 1, no. 2 (2013): 97–105. https://doi.org/10.5281/zenodo.3783841.

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A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of side length n for every n in the set. From [2] we know that N itself tiles the plane. From that and [3] we know that the set of even numbers tiles the plane while the set of odd numbers does not. According to [1] it is possible to tile the plane using only an odd square. In this paper we will check that if it is possible to tile the plane using an even set and n odd numbers or not.

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23

Abah, Joshua Abah. "Viewing basic math through the lens of history: Undergraduates' reflective learning in a history-augmented mathematics classroom." Waikato Journal of Education 22, no. 4 (2017): 33–48. https://doi.org/10.15663/wje.v22i4.557.

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This study is aimed at determining first-year university students’ reflections when Fibonacci tiling, the ancient Chinese fang cheng procedures, and the ancient Indian meru prastara recursions were introduced as historical snippets in an adventure pedagogy for basic mathematics. Seventy-eight firstyear students enrolled in a course in basic mathematics at a University in North Central Nigeria provided composite self-reports in an action research paradigm, describing their reflective learning after exposure to the historical snippets. Qualitative data reduction strategies were used to exp

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24

Zhuravlev, V. G. "One-dimensional Fibonacci tilings." Izvestiya: Mathematics 71, no. 2 (2007): 307–40. http://dx.doi.org/10.1070/im2007v071n02abeh002358.

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25

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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26

Baake, Michael, Franz Gähler, and Jan Mazáč. "Fibonacci direct product variation tilings." Journal of Mathematical Physics 63, no. 8 (2022): 082702. http://dx.doi.org/10.1063/5.0091099.

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The direct product of two Fibonacci tilings can be described as a genuine stone inflation rule with four prototiles. This rule admits various modifications, which lead to 48 different inflation rules, known as the direct product variations. They all result in tilings that are measure-theoretically isomorphic by the Halmos–von Neumann theorem. They can be described as cut and project sets with characteristic windows in a two-dimensional Euclidean internal space. Here, we analyze and classify them further, in particular, with respect to topological conjugacy.

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27

Treeby, David. "Hidden Formulas in Fibonacci Tilings." Fibonacci Quarterly 54, no. 1 (2016): 23–30. http://dx.doi.org/10.1080/00150517.2016.12427835.

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28

Kolá, M., and M. K. Ali. "One-dimensional generalized Fibonacci tilings." Physical Review B 41, no. 10 (1990): 7108–12. http://dx.doi.org/10.1103/physrevb.41.7108.

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29

Benjamin, Arthur T., Jennifer J. Quinn, and Francis Edward Su. "Phased Tilings and Generalized Fibonacci Identities." Fibonacci Quarterly 38, no. 3 (2000): 272–88. http://dx.doi.org/10.1080/00150517.2000.12428804.

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30

Margenstern, Maurice. "Fibonacci words, hyperbolic tilings and grossone." Communications in Nonlinear Science and Numerical Simulation 21, no. 1-3 (2015): 3–11. http://dx.doi.org/10.1016/j.cnsns.2014.07.032.

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31

Došlić, Tomislav, and Luka Podrug. "Tilings of a honeycomb strip and higher order Fibonacci numbers." Contributions to Discrete Mathematics 19, no. 2 (2024): 56–81. http://dx.doi.org/10.55016/ojs/cdm.v19i2.75062.

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In this paper we explore two types of tilings of a honeycomb strip and derive some closed form formulas for the number of tilings. Furthermore, we obtain some new identities involving tribonacci numbers, Padovan numbers and Narayana's cow sequence and provide combinatorial proofs for several known identities about those numbers.

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32

Neu, Keith, and Paul Deiermann. "Using Random Tilings to Derive a Fibonacci Congruence." College Mathematics Journal 37, no. 1 (2006): 44. http://dx.doi.org/10.2307/27646272.

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33

Nagy, Mariana, Simon R. Cowell, and Valeriu Beiu. "On the Construction of 3D Fibonacci Spirals." Mathematics 12, no. 2 (2024): 201. http://dx.doi.org/10.3390/math12020201.

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The paper aims to extend the classical two-dimensional (2D) Fibonacci spiral into three-dimensional (3D) space by using geometric constructions starting from cubic Fibonacci identities and relying on affine maps and parametrizations of the curves. We have already performed a comprehensive survey of cubic Fibonacci identities, which, to our surprise, uncovered only a handful of homogenous cubic identities. Obviously, the goal here is to show how one could use a particular homogenous cubic Fibonacci identity for generating 3D geometric designs similar in spirit to the way the classical Fibonacci

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34

Porrier, Carole. "The Leaf Function of Graphs Associated with Penrose Tilings." International Journal of Graph Computing 1, no. 1 (2020): 1–24. http://dx.doi.org/10.35708/gc1868-126721.

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In graph theory, the question of fully leafed induced subtrees has recently been investigated by Blondin Massé et al in regular tilings of the Euclidian plane and 3-dimensional space. The function LG that gives the maximum number of leaves of an induced subtree of a graph $G$ of order $n$, for any $n\in \N$, is called leaf function. This article is a first attempt at studying this problem in non-regular tilings, more specifically Penrose tilings. We rely not only on geometric properties of Penrose tilings, that allow us to find an upper bound for the leaf function in these tilings, but also on

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35

Dresden, Greg, and Yu Xiao. "Weighted Sums Of Fibonacci And Lucas Numbers Through Colorful Tilings." Fibonacci Quarterly 60, no. 2 (2022): 126–35. http://dx.doi.org/10.1080/00150517.2022.12427486.

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36

Németh, László. "Walks on tiled boards." Mathematica Slovaca 74, no. 6 (2024): 1369–82. https://doi.org/10.1515/ms-2024-0099.

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Abstract Several articles deal with tilings with various shapes, and also a very frequent type of combinatorics is to examine the walks on graphs or on grids. We combine these two things and give the numbers of the shortest walks crossing the tiled (1 × n) and (2 × n) square grids by covering them with squares and dominoes. We describe these numbers not only recursively, but also as rational polynomial linear combinations of Fibonacci numbers.

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37

Zhuravlev, Vladimir G. "One-dimensional Fibonacci tilings and induced two-colour rotations of the circle." Izvestiya: Mathematics 74, no. 2 (2010): 281–323. http://dx.doi.org/10.1070/im2010v074n02abeh002487.

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38

Holzer, Mark. "Three classes of one-dimensional, two-tile Penrose tilings and the Fibonacci Kronig-Penney model as a generic case." Physical Review B 38, no. 3 (1988): 1709–20. http://dx.doi.org/10.1103/physrevb.38.1709.

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39

Goy, Taras, and Mark Shattuck. "Toeplitz–Hessenberg determinant formulas for the sequence \(F_n-1\)." Online Journal of Analytic Combinatorics 19 (June 30, 2025): 1–26. https://doi.org/10.61091/ojac19-01.

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<p>Let <span class="math inline"><em>F</em><sub><em>n</em></sub></span> denote the <span class="math inline"><em>n</em></span>-th Fibonacci number defined by <span class="math inline"><em>F</em><sub><em>n</em></sub> = <em>F</em><sub><em>n</em> − 1</sub> + <em>F</em><sub><em>n</em> − 2</sub></span> if <span class="math inline"><em>n</em> ≥ 2</span>, with <span class="math i

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40

Abbott, Steve. "Charles Babbage: the man behind the machines, by Roger Webster. Mathematical Spectrum 24 (2) pp 34–41 - Finite Fibonacci Sequences, by I.M. Richards. Mathematical Spectrum 24 (2) pp 48–53 - On the optimum hand speed for two-blade razor shaving, by A.D. Fitt, A.A. Lacey and P. Wilmott. Teaching Mathematics and its Applications 10 (3) pp 122–126 - Nachruf: Theodor Schneider (1911-1988), by Peter Brundschuh and Hans Zassenhaus, Journal of Number Theory 39 (2) pp 129–143 - Tableaux de Young et Solitaire Bulgare, by Gwihen Etienne, Journal of Combinatorial Theory, Series A 58 (2) pp 181–197 - Rigor and proof in mathematics: a historical perspective, by Israel Kleiner Mathematics Magazine 64 (5) pp 291–311 - Reflections on a problem of V. Thébault, by H. Demu and C. Tezer, Geometriae Dedicata 39 (1) pp 79–92 - Some 2-isohedral tilings of the plane, by Richard L. Roth, Geometriae Dedicata 39 (1) pp 43–54 - An unorthodox “test”, by Abe Shenitzer, American Mathematical Monthly, 99 (1), pp 20–30." Mathematical Gazette 76, no. 477 (1992): 447–50. http://dx.doi.org/10.1017/s0025557200152180.

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41

Chandler, Liam, Oliver J. Barker, Alexander J. Wright, et al. "Fabricating Quasiperiodic Tilings with Thermal‐Scanning Probe Lithography." Israel Journal of Chemistry, December 11, 2023. http://dx.doi.org/10.1002/ijch.202300115.

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AbstractWe outline an approach to fabricate nanoscale artificial quasiperiodic tilings with thermal‐scanning probe lithography. Quasiperiodic tilings such as the Ammann‐Beenker, Square Fibonacci tiling, and Penrose are fabricated and imaged with thermal‐conductance feedback microscopy, followed by electron microscopy. The design implementation, chemical, and physical challenges involved in fabricating such artificial systems using nanolithography are discussed. Additionally, the potential applications of fabricated quasiperiodic tilings are explored.

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42

Khadir, Omar, László Németh, and László Szalay. "Tiling of dominoes with ranked colors." Results in Mathematics 79, no. 7 (2024). http://dx.doi.org/10.1007/s00025-024-02284-3.

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AbstractSeveral articles deal with tilings with various colors and shapes. In this paper, we present a new type of tiling problem of a $$(1\times n)$$ ( 1 × n ) —board where the colors have a prescribed order of preference and the size of colored dominoes is bounded by $$(1\times s)$$ ( 1 × s ) . We show that the total number of tilings can be given as a linearly recurrent sequence of order ks, and at the same time by a higher order self-convolution of s-generalized Fibonacci sequences.

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43

TAŞYURDU, Yasemin, and Naime Şeyda TÜRKOĞLU. "A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS." Journal of Universal Mathematics, July 30, 2022. http://dx.doi.org/10.33773/jum.1142805.

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In this paper, we introduce a tiling approach to (p,q)-Fibonacci and (p,q)-Lucas numbers that generalize of the well-known Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal ve Jacobsthal-Lucas numbers. We show that nth (p,q)-Fibonacci number is interpreted as the number of ways to tile a 1×n board with cells labeled 1,2,...,n using colored 1×1 squares and 1×2 dominoes, where there are p kind colors for squares and q kind colors for dominoes. Then nth (p,q)-Lucas number is interpreted as the number of ways to tile a circular 1×n board with squares and dominoes. We also present some generalized Fib

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44

TAŞYURDU, Yasemin, and Berke CENGİZ. "On Fibonaccı (k,p)-Numbers and Their Interpretations." Sakarya University Journal of Science, December 14, 2022. http://dx.doi.org/10.16984/saufenbilder.1173173.

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In this paper, we define a generalization of Fibonacci numbers, which generalizes both well-known Fibonacci, Jacobsthal, Narayana numbers and Fibonacci p-numbers in the distance sense, according to a new parameter k. Tiling and combinatorial interpretations of these generalized numbers are presented, and explicit formulas that allow us to calculate the 𝑛th number are given. Also, their generating functions and some properties are obtained.

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45

Coates, Sam, Akihisa Koga, Toranosuke Matsubara, et al. "Hexagonal and Trigonal Quasiperiodic Tilings." Israel Journal of Chemistry, September 5, 2024. http://dx.doi.org/10.1002/ijch.202300100.

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AbstractExploring nonminimal‐rank quasicrystals, which have symmetries that can be found in both periodic and aperiodic crystals, often provides new insight into the physical nature of aperiodic long‐range order in models that are easier to treat. Motivated by the prevalence of experimental systems exhibiting aperiodic long‐range order with hexagonal and trigonal symmetry, we introduce a generic two‐parameter family of 2‐dimensional quasiperiodic tilings with such symmetries. We focus on the special case of trigonal and hexagonal Fibonacci, or golden‐mean, tilings, analogous to the well studie

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46

Pouti, Aisling, and Nhi Phan. "Generating Functions Related to the Fibonacci Substitution." MacEwan University Student eJournal 7, no. 1 (2023). http://dx.doi.org/10.31542/muse.v7i1.2469.

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In this paper, two generating function representations of the Fibonacci Substitution Tiling are derived and proven to converge on the interval -1<x<1. A sequence of signs for the Fibonacci Substitution is established along with a conjecture that the interval of convergence has an infinite number of zeroes.

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47

Bodeen, John, Steve Butler, Taekyoung Kim, Xiyuan Sun, and Shenzhi Wang. "Tiling a Strip with Triangles." Electronic Journal of Combinatorics 21, no. 1 (2014). http://dx.doi.org/10.37236/3478.

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In this paper, we examine the tilings of a $2\times n$ "triangular strip" with triangles. These tilings have connections with Fibonacci numbers, Pell numbers, and other known sequences. We derive several different recurrences, establish some properties of these numbers, and give a refined count for these tilings (i.e., by the number and type of triangles used) and establish several properties of these refined counts.

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48

Eustis, Alex, and Mark Shattuck. "Combinatorial Proofs of Some Identities for the Fibonacci and Lucas Numbers." Integers 11, no. 5 (2011). http://dx.doi.org/10.1515/integ.2011.047.

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49

Baake, Michael, Franz Gähler, and Jan Mazáč. "On the Fibonacci Tiling and its Modern Ramifications." Israel Journal of Chemistry, April 12, 2024. http://dx.doi.org/10.1002/ijch.202300155.

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AbstractIn the last 30 years, the mathematical theory of aperiodic order has developed enormously. Many new tilings and properties have been discovered, few of which are covered or anticipated by the early papers and books. Here, we start from the well‐known Fibonacci chain to explain some of them, with pointers to various generalisations as well as to higher‐dimensional phenomena and results. This should give some entry points to the modern literature on the subject.

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50

Baake, Michael, Natalie Priebe Frank, and Uwe Grimm. "Three variations on a theme by Fibonacci." Stochastics and Dynamics, April 14, 2020, 2140001. http://dx.doi.org/10.1142/s0219493721400013.

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Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalization structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy frac

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