Fibonacci and Lucas Numbers with Applications

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John Wiley & Sons, 24 Eki 2011 - 672 sayfa
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The 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 Leonardo Fibonacci
1
2 The Rabbit Problem
4
3 Fibonacci Numbers in Nature
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
Index
641
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Yazar hakkında (2011)

THOMAS KOSHY, Ph.D., is Professor of Mathematics at Framingham State College in Framingham, Massachusetts. He is the author of the forthcoming book Elementary Number Theory with Applications, among several other books, as well as numerous articles across a wide range of topics.

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