*Thomas Koshy*

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

Fibonacci and Lucas sequences are “two shining stars in the vast array of integer sequences,” and because of their ubiquitousness, tendency to appear in quite unexpected and unrelated places, ...
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Fibonacci and Lucas sequences are “two shining stars in the vast array of integer sequences,” and because of their ubiquitousness, tendency to appear in quite unexpected and unrelated places, abundant applications, and intriguing properties, they have fascinated amateurs and mathematicians alike. However, Catalan numbers are even more fascinating. Like the North Star in the evening sky, they are a beautiful and bright light in the mathematical heavens. They continue to provide a fertile ground for number theorists, especially, Catalan enthusiasts and computer scientists. Since the publication of Euler's triangulation problem (1751) and Catalan's parenthesization problem (1838), over 400 articles and problems on Catalan numbers have appeared in various periodicals. As Martin Gardner noted, even though many amateurs and mathematicians may know the abc's of Catalan sequence, they may not be familiar with their myriad unexpected occurrences, delightful applications, properties, or the beautiful and surprising relationships among numerous examples. Like Fibonacci and Lucas numbers, Catalan numbers are also an excellent source of fun and excitement. They can be used to generate interesting dividends for students, such as intellectual curiosity, experimentation, pattern recognition, conjecturing, and problem-solving techniques. The central character in the nth Catalan number is the central binomial coefficient. So, Catalan numbers can be extracted from Pascal's triangle. In fact, there are a number of ways they can be read from Pascal's triangle; every one of them is described and exemplified. This brings Catalan numbers a step closer to number-theory enthusiasts, especially.Less

Fibonacci and Lucas sequences are “two shining stars in the vast array of integer sequences,” and because of their ubiquitousness, tendency to appear in quite unexpected and unrelated places, abundant applications, and intriguing properties, they have fascinated amateurs and mathematicians alike. However, Catalan numbers are even more fascinating. Like the North Star in the evening sky, they are a beautiful and bright light in the mathematical heavens. They continue to provide a fertile ground for number theorists, especially, Catalan enthusiasts and computer scientists. Since the publication of Euler's triangulation problem (1751) and Catalan's parenthesization problem (1838), over 400 articles and problems on Catalan numbers have appeared in various periodicals. As Martin Gardner noted, even though many amateurs and mathematicians may know the *abc*'s of Catalan sequence, they may not be familiar with their myriad unexpected occurrences, delightful applications, properties, or the beautiful and surprising relationships among numerous examples. Like Fibonacci and Lucas numbers, Catalan numbers are also an excellent source of fun and excitement. They can be used to generate interesting dividends for students, such as intellectual curiosity, experimentation, pattern recognition, conjecturing, and problem-solving techniques. The central character in the *n*th Catalan number is the central binomial coefficient. So, Catalan numbers can be extracted from Pascal's triangle. In fact, there are a number of ways they can be read from Pascal's triangle; every one of them is described and exemplified. This brings Catalan numbers a step closer to number-theory enthusiasts, especially.

*Thomas Koshy*

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0003
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter continues to investigate the central binomial coefficient in much greater detail. Topics explained include Wallis' product, Lucas numbers, Fermat's little theorem, Carlitz's formula, ...
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This chapter continues to investigate the central binomial coefficient in much greater detail. Topics explained include Wallis' product, Lucas numbers, Fermat's little theorem, Carlitz's formula, Stirling's asymptotic formula, Fibonacci numbers, Lucas' formula, and Norton's formula.Less

This chapter continues to investigate the central binomial coefficient in much greater detail. Topics explained include Wallis' product, Lucas numbers, Fermat's little theorem, Carlitz's formula, Stirling's asymptotic formula, Fibonacci numbers, Lucas' formula, and Norton's formula.

*Susan D'Agostino*

- Published in print:
- 2020
- Published Online:
- April 2020
- ISBN:
- 9780198843597
- eISBN:
- 9780191879388
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198843597.003.0014
- Subject:
- Mathematics, Educational Mathematics, Applied Mathematics

“Look for beauty, because of Fibonacci numbers” explains how to use addition to construct an ordered list of numbers known as the Fibonacci numbers. The chapter explores several unexpected and lovely ...
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“Look for beauty, because of Fibonacci numbers” explains how to use addition to construct an ordered list of numbers known as the Fibonacci numbers. The chapter explores several unexpected and lovely examples of these numbers in nature, including the numbers of clockwise and counterclockwise spirals in sunflowers, cacti, and pinecones. The discussion includes numerous photographs illustrating Fibonacci numbers in nature. Mathematics students and enthusiasts are encouraged to accept the daily invitation to practice searching for and finding abundant mathematical beauty around them. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.Less

“Look for beauty, because of Fibonacci numbers” explains how to use addition to construct an ordered list of numbers known as the Fibonacci numbers. The chapter explores several unexpected and lovely examples of these numbers in nature, including the numbers of clockwise and counterclockwise spirals in sunflowers, cacti, and pinecones. The discussion includes numerous photographs illustrating Fibonacci numbers in nature. Mathematics students and enthusiasts are encouraged to accept the daily invitation to practice searching for and finding abundant mathematical beauty around them. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

*A. W. F. EDWARDS*

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199656592
- eISBN:
- 9780191748059
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199656592.003.0008
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics

The arithmetical triangle is the most famous of all number patterns. Apparently a simple listing of the binomial coefficients, it contains the triangular and pyramidal numbers of ancient Greece, the ...
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The arithmetical triangle is the most famous of all number patterns. Apparently a simple listing of the binomial coefficients, it contains the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers that arose in the Hindu studies of arrangements and selections, and (barely concealed) the Fibonacci numbers from medieval Italy. It reveals patterns that delight the eye, raises questions that tax the number-theorists, and amongst the ‘… coefficients, there are so many relations present that when someone finds a new identity, there aren’t many people who get excited about it any more, except the discoverer!’Less

The arithmetical triangle is the most famous of all number patterns. Apparently a simple listing of the binomial coefficients, it contains the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers that arose in the Hindu studies of arrangements and selections, and (barely concealed) the Fibonacci numbers from medieval Italy. It reveals patterns that delight the eye, raises questions that tax the number-theorists, and amongst the ‘… coefficients, there are so many relations present that when someone finds a new identity, there aren’t many people who get excited about it any more, except the discoverer!’