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<p><b>1 Leonardo Fibonacci 9</b></p> <p><b>2 Fibonacci Numbers 13</b></p> <p>2.1 Fibonacci's Rabbits 13</p> <p>2.2 Fibonacci Numbers 14</p> <p>2.3 Fibonacci and Lucas Curiosities 17</p> <p><b>3 Fibonacci Numbers in Nature 27</b></p> <p>3.1 Fibonacci, Flowers, and Trees 28</p> <p>3.2 Fibonacci and Male Bees 31</p> <p>3.3 Fibonacci, Lucas, and Subsets 32</p> <p>3.4 Fibonacci and Sewage Treatment 34</p> <p>3.5 Fibonacci and Atoms 35</p> <p>3.6 Fibonacci and Reflections 36</p> <p>3.7 Paraffins and Cycloparaffins 38</p> <p>3.8 Fibonacci and Music 41</p> <p>3.9 Fibonacci and Poetry 42</p> <p>3.10 Fibonacci and Neurophysiology 43</p> <p>3.11 Electrical Networks 45</p> <p><b>4 Additional Fibonacci and Lucas Occurrences 53</b></p> <p>4.1 Fibonacci Occurrences 53</p> <p>4.2 Fibonacci and Compositions 58</p> <p>4.3 Fibonacci and Permutations 61</p> <p>4.4 Fibonacci and Generating Sets 63</p> <p>4.5 Fibonacci and Graph Theory 64</p> <p>4.6 Fibonacci Walks 66</p> <p>4.7 Fibonacci Trees 68</p> <p>4.8 Partitions 71</p> <p>4.9 Fibonacci and the Stock Market 72</p> <p><b>5 Fibonacci and Lucas Identities 77</b></p> <p>5.1 Spanning Tree of a Connected Graph 79</p> <p>5.2 Binet's Formulas 83</p> <p>5.3 Cyclic Permutations and Lucas Numbers 91</p> <p>5.4 Compositions Revisited 94</p> <p>5.5 Number of Digits in Fn and Ln 94</p> <p>5.6 Theorem 5.8 Revisited 95</p> <p>5.7 Catalan's Identity 99</p> <p>5.8 Additional Fibonacci and Lucas Identities 102</p> <p>5.9 Fermat and Fibonacci 108</p> <p>5.10 Fibonacci and  110</p> <p><b>6 Geometric Illustrations and Paradoxes 117</b></p> <p>6.1 Geometric Illustrations 117</p> <p>6.2 Candido's Identity 121</p> <p>6.3 Fibonacci Tessellations 123</p> <p>6.4 Lucas Tessellations 123</p> <p>6.5 Geometric Paradoxes 124</p> <p>6.6 Cassini-Based Paradoxes 124</p> <p>6.7 Additional Paradoxes 129</p> <p><b>7 Gibonacci Numbers 133</b></p> <p>7.1 Gibonacci Numbers 133</p> <p>7.2 Germain's Identity 139</p> <p><b>8 Additional Fibonacci and Lucas Formulas 145</b></p> <p>8.1 New Explicit Formulas 145</p> <p>8.2 Additional Formulas 148</p> <p><b>9 The Euclidean Algorithm 159</b></p> <p>9.1 The Euclidean Algorithm 160</p> <p>9.2 Formula (5.5) Revisited 162</p> <p>9.3 Lamé's Theorem 164</p> <p><b>10 Divisibility Properties 167</b></p> <p>10.1 Fibonacci Divisibility 167</p> <p>10.2 Lucas Divisibility 173</p> <p>10.3 Fibonacci and Lucas Ratios 173</p> <p>10.4 An Altered Fibonacci Sequence 178</p> <p><b>11 Pascal's Triangle 185</b></p> <p>11.1 Binomial Coefficients 185</p> <p>11.2 Pascal's Triangle 186</p> <p>11.3 Fibonacci Numbers and Pascal’s Triangle 188</p> <p>11.4 Another Explicit Formula for Ln 191</p> <p>11.5 Catalan's Formula 192</p> <p>11.6 Additional Identities 192</p> <p>11.7 Fibonacci Paths of a Rook on a Chessboard 194</p> <p><b>12 Pascal-like Triangles 199</b></p> <p>12.1 Sums of Like-Powers 199</p> <p>12.2 An Alternate Formula for Ln 202</p> <p>12.3 Differences of Like-Powers 202</p> <p>12.4 Catalan's Formula Revisited 204</p> <p>12.5 A Lucas Triangle 205</p> <p>12.6 Powers of Lucas Numbers 209</p> <p>12.7 Variants of Pascal's Triangle 211</p> <p><b>13 Recurrences and Generating Functions 219</b></p> <p>13.1 LHRWCCs 219</p> <p>13.2 Generating Functions 223</p> <p>13.3 A Generating Function For F3n 233</p> <p>13.4 A Generating Function For F3 n 234</p> <p>13.5 Summation Formula (5.1) Revisited 234</p> <p>13.6 A List of Generating Functions 235</p> <p>13.7 Compositions Revisited 238</p> <p>13.8 Exponential Generating Functions 239</p> <p>13.9 Hybrid Identities 241</p> <p>13.10Identities Using the Differential Operator 242</p> <p><b>14 Combinatorial Models I 249</b></p> <p>14.1 A Fibonacci Tiling Model 249</p> <p>14.2 A Circular Tiling Model 255</p> <p>14.3 Path Graphs Revisited 259</p> <p>14.4 Cycle Graphs Revisited 262</p> <p>14.5 Tadpole Graphs 263</p> <p><b>15 Hosoya's Triangle 271</b></p> <p>15.1 Recursive Definition 271</p> <p>15.2 A Magic Rhombus 273</p> <p><b>16 The Golden Ratio 279</b></p> <p>16.1 Ratios of Consecutive Fibonacci Numbers 279</p> <p>16.2 The Golden Ratio 281</p> <p>16.3 Golden Ratio as Nested Radicals 285</p> <p>16.4 Newton's Approximation Method 286</p> <p>16.5 The Ubiquitous Golden Ratio 288</p> <p>16.6 Human Body and the Golden Ratio 289</p> <p>16.7 Violin and the Golden Ratio 290</p> <p>16.8 Ancient Floor Mosaics and the Golden Ratio 290</p> <p>16.9 Golden Ratio in an Electrical Network 290</p> <p>16.10Golden Ratio in Electrostatics 291</p> <p>16.11Golden Ratio by Origami 292</p> <p>16.12Differential Equations 297</p> <p>16.13Golden Ratio in Algebra 299</p> <p>16.14Golden Ratio in Geometry 300</p> <p><b>17 Golden Triangles and Rectangles 309</b></p> <p>17.1 Golden Triangle 309</p> <p>17.2 Golden Rectangles 314</p> <p>17.3 The Parthenon 317</p> <p>17.4 Human Body and the Golden Rectangle 318</p> <p>17.5 Golden Rectangle and the Clock 319</p> <p>17.6 Straightedge and Compass Construction 320</p> <p>17.7 Reciprocal of a Rectangle 321</p> <p>17.8 Logarithmic Spiral 322</p> <p>17.9 Golden Rectangle Revisited 324</p> <p>17.10Supergolden Rectangle 324</p> <p><b>18 Figeometry 329</b></p> <p>18.1 The Golden Ratio and Plane Geometry 329</p> <p>18.2 The Cross of Lorraine 335</p> <p>18.3 Fibonacci Meets Appollonius 337</p> <p>18.4 A Fibonacci Spiral 338</p> <p>18.5 Regular Pentagons 339</p> <p>18.6 Trigonometric Formulas for Fn 343</p> <p>18.7 Regular Decagon 347</p> <p>18.8 Fifth Roots of Unity 348</p> <p>18.9 A Pentagonal Arch 351</p> <p>18.10 Regular Icosahedron and Dodecahedron 351</p> <p>18.11 Golden Ellipse 352</p> <p>18.12 Golden Hyperbola 354</p> <p><b>19 Continued Fractions 361</b></p> <p>19.1 Finite Continued Fractions 361</p> <p>19.2 Convergents of a Continued Fraction 364</p> <p>19.3 Infinite Continued Fractions 366</p> <p>19.4 A Nonlinear Diophantine Equation 368</p> <p><b>20 Fibonacci Matrices 371</b></p> <p>20.1 The Q-Matrix 371</p> <p>20.2 Eigenvalues of Qn 378</p> <p>20.3 Fibonacci and Lucas Vectors 384</p> <p>20.4 An Intriguing Fibonacci Matrix 386</p> <p>20.5 An Infinite-Dimensional Lucas Matrix 391</p> <p>20.6 An Infinite-Dimensional Gibonacci Matrix 397</p> <p>20.7 The Lambda Function 398</p> <p><b>21 Graph-theoretic Models I 407</b></p> <p>21.1 A Graph-theoretic Model for Fibonacci Numbers 407</p> <p>21.2 Byproducts of the Combinatorial Models 409</p> <p>21.3 Summation Formulas 415</p> <p><b>22 Fibonacci Determinants 419</b></p> <p>22.1 An Application to Graph Theory 419</p> <p>22.2 The Singularity of Fibonacci Matrices 425</p> <p>22.3 Fibonacci and Analytic Geometry 427</p> <p><b>23 Fibonacci and Lucas Congruences 437</b></p> <p>23.1 Fibonacci Numbers Ending in Zero 437</p> <p>23.2 Lucas Numbers Ending in Zero 437</p> <p>23.3 Additional Congruences 438</p> <p>23.4 Lucas Squares 439</p> <p>23.5 Fibonacci Squares 440</p> <p>23.6 A Generalized Fibonacci Congruence 442</p> <p>23.7 Fibonacci and Lucas Periodicities 449</p> <p>23.8 Lucas Squares Revisited 450</p> <p>23.9 Periodicities Modulo 10n 452</p> <p><b>24 Fibonacci and Lucas Series 461</b></p> <p>24.1 A Fibonacci Series 461</p> <p>24.2 A Lucas Series 463</p> <p>24.3 Fibonacci and Lucas Series Revisited 464</p> <p>24.4 A Fibonacci Power Series 467</p> <p>24.5 Gibonacci Series 472</p> <p>24.6 Additional Fibonacci Series 474</p> <p><b>25 Weighted Fibonacci and Lucas Sums 481</b></p> <p>25.1 Weighted Sums 481</p> <p>25.2 Gauthier's Differential Method 488</p> <p><b>26 Fibonometry I 495</b></p> <p>26.1 Golden Ratio and Inverse Trigonometric Functions 495</p> <p>26.2 Golden Triangle Revisited 496</p> <p>26.3 Golden Weaves 497</p> <p>26.4 Additional Fibonometric Bridges 498</p> <p>26.5 Fibonacci and Lucas Factorizations 504</p> <p><b>27 Completeness Theorems 509</b></p> <p>27.1 Completeness Theorem 509</p> <p>27.2 Egyptian Algorithm for Multiplication 510</p> <p><b>28 The Knapsack Problem 513</b></p> <p>28.1 The Knapsack Problem 513</p> <p><b>29 Fibonacci and Lucas Subscripts 517</b></p> <p>29.1 Fibonacci and Lucas Subscripts 517</p> <p>29.2 Gibonacci Subscripts 519</p> <p>29.3 A Recursive Definition of Yn 520</p> <p><b>30 Fibonacci and the Complex Plane 525</b></p> <p>30.1 Gaussian Numbers 525</p> <p>30.2 Gaussian Fibonacci and Lucas Numbers 526</p> <p>30.3 Analytic Extensions 530</p> <p>1 A.1 Fundamentals 537</p> <p>SOLUTIONS TO ODD-NUMBERED EXERCISES 575</p>

<p><b> Thomas Koshy, PhD,</b> is Professor Emeritus of Mathematics at Framingham State University in Massachusetts and author of several books and numerous articles on mathematics. His work has been recognized by the Association of American Publishers, and he has received many awards, including the Distinguished Faculty of the Year. Dr. Koshy received his PhD in Algebraic Coding Theory from Boston University.

<p><b> Praise for the First Edition</b><br> "...beautiful and well worth the reading ... with many exercises and a good bibliography, this book will fascinate both students and teachers."<br> <b><i>Mathematics Teacher </i></b> <p><i> Fibonacci and Lucas Numbers with Applications, Volume I, Second Edition</i> provides a user-friendly and historical approach to the many fascinating properties of Fibonacci and Lucas numbers, which have intrigued amateurs and professionals for centuries. Offering an in-depth study of the topic, this book includes exciting applications that provide many opportunities to explore and experiment. <p> In addition, the book includes a historical survey of the development of Fibonacci and Lucas numbers, with biographical sketches of important figures in the field. Each chapter features a wealth of examples, as well as numeric and theoretical exercises that avoid using extensive and time-consuming proofs of theorems. The <i>Second Edition</i> offers new opportunities to illustrate and expand on various problem-solving skills and techniques. In addition, the book features: <ul> <li>A clear, comprehensive introduction to one of the most fascinating topics in mathematics, including links to graph theory, matrices, geometry, the stock market, and the Golden Ratio</li> <li>Abundant examples, exercises, and properties throughout, with a wide range of difficulty and sophistication</li> <li>Numeric puzzles based on Fibonacci numbers, as well as popular geometric paradoxes, and a glossary of symbols and fundamental properties from the theory of numbers</li> <li>A wide range of applications in many disciplines, including architecture, biology, chemistry, electrical engineering, physics, physiology, and neurophysiology</li> </ul> <br> <p> The <i>Second Edition</i> is appropriate for upper-undergraduate and graduate-level courses on the history of mathematics, combinatorics, and number theory. The book is also a valuable resource for undergraduate research courses, independent study projects, and senior/graduate theses, as well as a useful resource for computer scientists, physicists, biologists, and electrical engineers. <p> "Anyone who loves mathematical puzzles, number theory, and Fibonacci numbers will treasure this book. Dr. Koshy has compiled Fibonacci lore from diverse sources into one understandable and intriguing volume, [interweaving] a historical flavor into an array of applications."<br> <b><i>Marjorie Bicknell-Johnson </i></b>

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