List of Symbols xiii ; Preface xv ; 31. Fibonacci and Lucas Polynomials I 1 ; 31.1. Fibonacci and Lucas Polynomials 3 ; 31.2. Pascal's Triangle 18 ; 31.3. Additional Explicit Formulas 22 ; 31.4. Ends of the Numbers l_n 25 ; 31.5. Generating Functions 26 ; 31.6. Pell and Pell-Lucas Polynomials 27 ; 31.7. Composition of Lucas Polynomials 33 ; 31.8. De Moivre-like Formulas 35 ; 31.9. Fibonacci-Lucas Bridges 36 ; 31.10. Applications of Identity (31.51) 37 ; 31.11. Infinite Products 48 ; 31.12. Putnam Delight Revisited 51 ; 31.13. Infinite Simple Continued Fraction 54 ; 32. Fibonacci and Lucas Polynomials II 65 ; 32.1. Q-Matrix 65 ; 32.2. Summation Formulas 67 ; 32.3. Addition Formulas 71 ; 32.4. A Recurrence for f_n² 76 ; 32.5. Divisibility Properties 82 ; 33. Combinatorial Models II 87 ; 33.1. A Model for Fibonacci Polynomials 87 ; 33.2. Breakability 99 ; 33.3. A Ladder Model 101 ; 33.4. A Model for Pell-Lucas Polynomials: Linear Boards 102 ; 33.5. Colored Tilings 103 ; 33.6. A New Tiling Scheme 104 ; 33.7. A Model for Pell-Lucas Polynomials: Circular Boards 107 ; 33.8. A Domino Model for Fibonacci Polynomials 114 ; 33.9. Another Model for Fibonacci Polynomials 118 ; 34. Graph-Theoretic Models II 125 ; 34.1. Q-Matrix and Connected Graph 125 ; 34.2. Weighted Paths 126 ; 34.3. Q-Matrix Revisited 127 ; 34.4. Byproducts of the Model 128 ; 34.5. A Bijection Algorithm 136 ; 34.6. Fibonacci and Lucas Sums 137 ; 34.7. Fibonacci Walks 140 ; 35. Gibonacci Polynomials 145 ; 35.1. Gibonacci Polynomials 145 ; 35.2. Differences of Gibonacci Products 159 ; 35.3. Generalized Lucas and Ginsburg Identities 174 ; 35.4. Gibonacci and Geometry 181 ; 35.5. Additional Recurrences 184 ; 35.6. Pythagorean Triples 188 ; 36. Gibonacci Sums 195 ; 36.1. Gibonacci Sums 195 ; 36.2. Weighted Sums 206 ; 36.3. Exponential Generating Functions 209 ; 36.4. Infinite Gibonacci Sums 215 ; 37. Additional Gibonacci Delights 233 ; 37.1. Some Fundamental Identities Revisited 233 ; 37.2. Lucas and Ginsburg Identities Revisited 238 ; 37.3. Fibonomial Coefficients 247 ; 37.4. Gibonomial Coefficients 250 ; 37.5. Additional Identities 260 ; 37.6. Strazdins' Identity 264 ; 38. Fibonacci and Lucas Polynomials III 269 ; 38.1. Seiffert's Formulas 270 ; 38.2. Additional Formulas 294 ; 38.3. Legendre Polynomials 314 ; 39. Gibonacci Determinants 321 ; 39.1. A Circulant Determinant 321 ; 39.2. A Hybrid Determinant 323 ; 39.3. Basin's Determinant 333 ; 39.4. Lower Hessenberg Matrices 339 ; 39.5. Determinant with a Prescribed First Row 343 ; 40. Fibonometry II 347 ; 40.1. Fibonometric Results 347 ; 40.2. Hyperbolic Functions 356 ; 40.3. Inverse Hyperbolic Summation Formulas 361 ; 41. Chebyshev Polynomials 371 ; 41.1. Chebyshev Polynomials T_n(x) 372 ; 41.2. T_n(x) and Trigonometry 384 ; 41.3. Hidden Treasures in Table 41.1 386 ; 41.4. Chebyshev Polynomials U_n(x) 396 ; 41.5. Pell's Equation 398 ; 41.6. U n (x) and Trigonometry 399 41.7. Addition and Cassini-like Formulas 401 41.8. Hidden Treasures in Table 41.8 402 41.9. A Chebyshev Bridge 404 ; 41.10. T_n and U_n as Products 405 ; 41.11. Generating Functions 410 42. Chebyshev Tilings 415 42.1. Combinatorial Models for U 415 42.2. Combinatorial Models for T 420 42.3. Circular Tilings 425 43. Bivariate Gibonacci Family I 429 ; 43.1. Bivariate Gibonacci Polynomials 429 ; 43.2. Bivariate Fibonacci and Lucas Identities 430 ; 43.3. Candido's Identity Revisited 439 ; 44. Jacobsthal Family 443 ; 44.1. Jacobsthal Family 444 ; 44.2. Jacobsthal Occurrences 450 ; 44.3. Jacobsthal Compositions 452 ; 44.4. Triangular Numbers in the Family 459 ; 44.5. Formal Languages 468 ; 44.6. A USA Olympiad Delight 480 ; 44.7. A Story of 1, 2, 7, 42, 429,...483 ; 44.8. Convolutions 490 ; 45. Jacobsthal Tilings and Graphs 499 ; 45.1. 1 × n Tilings 499 ; 45.2. 2 × n Tilings 505 ; 45.3. 2 × n Tubular Tilings 510 ; 45.4. 3 × n Tilings 514 ; 45.5. Graph-Theoretic Models 518 ; 45.6. Digraph Models 522 ; 46. Bivariate Tiling Models 537 ; 46.1. A Model for 537 ; 46.2. Breakability 539 ; 46.3. Colored Tilings 542 ; 46.4. A Model for l x y 543 46.5. Colored Tilings Revisited 545 ; 46.6. Circular Tilings Again 547 ; 47. Vieta Polynomials 553 ; 47.1. Vieta Polynomials 554 ; 47.2. Aurifeuille's Identity 567 ; 47.3. Vieta-Chebyshev Bridges 572 ; 47.4. Jacobsthal-Chebyshev Links 573 ; 47.5. Two Charming Vieta Identities 574 47.6. Tiling Models for V n 576 47.7. Tiling Models for 48. Bivariate Gibonacci Family II 591 48.1. Bivariate Identities 591 48.2. Additional Bivariate Identities 594 ; 48.3. A Bivariate Lucas Counterpart 599 ; 48.4. A Summation Formula for 600 ; 48.5. A Summation Formula for l_2n(x, y) 602 ; 48.6. Bivariate Fibonacci Links 603 ; 48.7. Bivariate Lucas Links 606 ; 49. Tribonacci Polynomials 611 ; 49.1. Tribonacci Numbers 611 ; 49.2. Compositions with Summands 1, 2, and 3 613 ; 49.3. Tribonacci Polynomials 616 ; 49.4. A Combinatorial Model 618 ; 49.5. Tribonacci Polynomials and the Q-Matrix 624 ; 49.6. Tribonacci Walks 625 ; 49.7. A Bijection between the Two Models 627 ; Appendix 631 ; A.1. The First 100 Fibonacci and Lucas Numbers 631 ; A.2. The First 100 Pell and Pell-Lucas Numbers 634 ; A.3. The First 100 Jacobsthal and Jacobsthal-Lucas Numbers 638 ; A.4. The First 100 Tribonacci Numbers 642 ; Abbreviations 644 ; Bibliography 645 ; Solutions to Odd-Numbered Exercises 661 ; Index 725