Fibonacci and Lucas Numbers with Applications

676 Pages·2001·16.92 MB·English

by  Thomas Koshy

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FIBONACCI AND LUCAS
NUMBERS WITH
APPLICATIONS
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editors: MYRON B. ALLEN III, DAVID A. COX, PETER LAX
Editors Emeriti: PETER HILTON, HARRY HOCHSTADT, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
FIBONACCI AND LUCAS
NUMBERS WITH
APPLICATIONS
THOMAS KOSHY
Framingham State College
A Wiley-Interscience Publication
JOHN WILEY & SONS, INC.
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This book is printed on acid-free paper. ®
Copyright © 2001 by John Wiley & Sons, Inc. All rights reserved.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form
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Library of Congress Cataloging-in-Publication Data:
Koshy, Thomas.
Fibonacci and Lucas numbers with applications/Thomas Koshy.
p. cm. — (Pure and applied mathematics: a Wiley-Interscience series of texts,
monographs, and tracts)
"A Wiley-Interscience publication."
Includes bibliographical references and index.
ISBN 0-471-39969-8 (cloth : alk. paper)
1. Fibonacci numbers. 2. Lucas numbers. I. Title II. Pure and applied mathematics
(John Wiley & Sons: Unnumbered)
QA246.5.K67 2001
5I2'.72—dc21 2001017506
10 987654321
To
Suresh, Neethu, and Sheeba
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CONTENTS
Preface xi
List of Symbols xv
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
4
9. The Euclidean Algorithm 132
10. Solving Recurrence Relations 142
11. Completeness Theorems 147
12. Pascal's Triangle 151
13. Pascal-Like Triangles 164
14. Additional Pascal-Like Triangles 180
15. Hosoya's Triangle 187
16. Divisibility Properties 196
vii
viii
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
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
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. Morgan- Voyce 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