## Fibonacci and Lucas Numbers with ApplicationsThe first comprehensive survey of mathematics' most fascinating number sequences Fibonacci and Lucas numbers have intrigued amateur and professional mathematicians for centuries. This volume represents the first attempt to compile a definitive history and authoritative analysis of these famous integer sequences, complete with a wealth of exciting applications, enlightening examples, and fun exercises that offer numerous opportunities for exploration and experimentation. The author has assembled a myriad of fascinating properties of both Fibonacci and Lucas numbers-as developed by a wide range of sources-and catalogued their applications in a multitude of widely varied disciplines such as art, stock market investing, engineering, and neurophysiology. Most of the engaging and delightful material here is easily accessible to college and even high school students, though advanced material is included to challenge more sophisticated Fibonacci enthusiasts. A historical survey of the development of Fibonacci and Lucas numbers, biographical sketches of intriguing personalities involved in developing the subject, and illustrative examples round out this thorough and amusing survey. Most chapters conclude with numeric and theoretical exercises that do not rely on long and tedious proofs of theorems. Highlights include: * Balanced blend of theory and real-world applications * Excellent reference material for student reports and projects * User-friendly, informal, and entertaining writing style * Historical interjections and short biographies that add a richer perspective to the topic * Reference sections providing important symbols, problem solutions, and fundamental properties from the theory of numbers and matrices Fibonacci and Lucas Numbers with Applications provides mathematicians with a wealth of reference material in one convenient volume and presents an in-depth and entertaining resource for enthusiasts at every level and from any background. |

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### İçindekiler

1 | |

4 | |

16 | |

4 Fibonacci Numbers Additional Occurrences | 51 |

5 Fibonacci and Lucas Identities | 69 |

6 Geometric Paradoxes | 100 |

7 Generalized Fibonacci Numbers | 109 |

8 Additional Fibonacci and Lucas Formulas | 116 |

27 Continued Fractions | 332 |

28 Weighted Fibonacci and Lucas Sums | 340 |

29 Fibonacci and Lucas Sums Revisited | 349 |

30 The Knapsack Problem | 356 |

31 Fibonacci Magic Squares | 360 |

32 Fibonacci Matrices | 362 |

33 Fibonacci Determinants | 387 |

34 Fibonacci and Lucas Congruences | 402 |

9 The Euclidean Algorithm | 132 |

10 Solving Recurrence Relations | 142 |

11 Completeness Theorems | 147 |

12 Pascals Triangle | 151 |

13 PascalLike Triangles | 164 |

14 Additional PascalLike Triangles | 180 |

15 Hosoyas Triangle | 187 |

16 Divisibility Properties | 196 |

17 Generalized Fibonacci Numbers Revisited | 211 |

18 Generating Functions | 215 |

19 Generating Functions Revisited | 227 |

20 The Golden Ratio | 239 |

21 The Golden Ratio Revisited | 248 |

22 Golden Triangles | 267 |

23 Golden Rectangles | 273 |

24 Fibonacci Geometry | 294 |

25 Regular Pentagons | 308 |

26 The Golden Ellipse and Hyperbola | 328 |

35 Fibonacci and Lucas Periodicity | 415 |

36 Fibonacci and Lucas Series | 424 |

37 Fibonacci Polynomials | 443 |

38 Lucas Polynomials | 459 |

39 Jacobsthal Polynomials | 469 |

40 Zeros of Fibonacci and Lucas Polynomials | 477 |

41 MorganVoyce Polynomials | 480 |

42 Fibonometry | 496 |

43 Fibonacci and Lucas Subscripts | 511 |

44 Gaussian Fibonacci and Lucas Numbers | 518 |

45 Analytic Extensions | 523 |

46 Tribonacci Numbers | 527 |

47 Tribonacci Polynomials | 533 |

Appendix | 537 |

References | 562 |

Solutions to OddNumbered Exercises | 577 |

641 | |

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AABC April Bicknell Binet’s formula Carlitz Cassini’s formula Chapter coefﬁcients compute conﬁrm consecutive Fibonacci numbers continued fraction converges Corollary deﬁned denote the number digits equation Euclidean algorithm example Exercise explicit formula F2n+l Fibonacci and Lucas Fibonacci numbers Fibonacci polynomials Fibonacci Quarterly Fibonacci recurrence relation Fibonacci sequence Fibonacci tree Figure ﬁnd ﬁnite ﬁrst ﬁve Fn—l Fn+1 Fn+2 Fn+l following result function geometry Golden Ratio golden rectangle golden triangle graph Identity inﬁnite knapsack problem Koshy Lemma Lucas numbers Lucas sequence Mathematical Gazette mathematical induction matrix Notice Pascal’s triangle pentagon perfect square positive integer prime Proof properties Prove recurrence relation recursive deﬁnition reﬂection result is true rising diagonal satisﬁes side Solution to Problem subsets Suppose Swamy Table tree tribonacci V. E. Hoggatt values Verify vertex vertices yields zero