Fibonacci and Lucas Numbers with Applications, Volume 1

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