Fibonacci Identities Research Articles

Set partition statistics and [formula omitted]-Fibonacci numbers

We consider the set partition statistics l s and r b introduced by Wachs and White and investigate their distribution over set partitions that avoid certain patterns. In particular, we consider those set partitions avoiding the pattern 13 / 2 , Π n ( 13 / 2 ) , and those avoiding both 13/2 and 123, Π n ( 13 / 2 , 123 ) . We show that the distribution over Π n ( 13 / 2 ) enumerates certain integer partitions, and the distribution over Π n ( 13 / 2 , 123 ) gives q -Fibonacci numbers. These q -Fibonacci numbers are closely related to q -Fibonacci numbers studied by Carlitz and by Cigler. We provide combinatorial proofs that these q -Fibonacci numbers satisfy q -analogues of many Fibonacci identities. Finally, we indicate how p , q -Fibonacci numbers arising from the bistatistic ( l s , r b ) give rise to p , q -analogues of identities.

Exploring recursion with fibonacci numbers

This paper addresses the relationship between recursion and combinatorial problems, which may benefit teaching recursion in CS1/2 courses. Problems whose solutions are related to Fibonacci numbers are particularly interesting, since some can be decomposed by using different strategies, which may be based on the numerous Fibonacci identities or even on the concept of mutual recursion.

Bijective Proofs for Fibonacci Identities Related to Zeckendorf’s Theorem

Bijective Proofs for Fibonacci Identities Related to Zeckendorf’s Theorem

On computing the determinants and inverses of some special type of tridiagonal and constant-diagonals matrices

The determinants and inverses of some tridiagonal and constant-diagonals matrices are discussed in the paper. Linear recurrence equations of the second and higher orders which satisfy these determinants are presented here. In connection with these determinants new families of polynomials are defined. The relationships between these polynomials and normed Chebyshev polynomials of the first and second kind are investigated. Incidentally, a number of new identities for Chebyshev polynomials and some linear modifications of these polynomials are shown, as well as many new trigonometric identities and identities for Fibonacci and Lucas numbers.

Fibonacci Identities via the Determinant Sum Property

(2006). Fibonacci Identities via the Determinant Sum Property. The College Mathematics Journal: Vol. 37, No. 4, pp. 286-289.

Product Difference Fibonacci Identities of Simson, Gelin-Cesaro, Tagiuri and Generalizations

Product Difference Fibonacci Identities of Simson, Gelin-Cesaro, Tagiuri and Generalizations

Delving Deeper: Fibonacci Identities, Matrices, and Graphs

In “Fibonacci and Related Sequences” published in the “Delving Deeper” department of the Mathematics Teacher February 2004, Richard Askey presents “general strategies used to help discover, prove, and generalize” identities for Fibonacci numbers. He also suggests some problems for further delving, including proving one key identity by using matrices. That proof is the focus of this article.

A High-Powered Fibonacci Identity: 10774

A High-Powered Fibonacci Identity: 10774

Random Approaches to Fibonacci Identities

Many combinatorialists live by Mach’s words, and take it as a personal challenge. For example, nearly all of the Fibonacci identities in [5] and [6] have been explained by counting arguments [1, 2, 3]. Among the holdouts are those involving infinite sums and irrational quantities. However, by adopting a probabilistic viewpoint, many of the remaining identities can be explained combinatorially. As we shall demonstrate, even the “irrational-looking” Binet’s formula for the n-th Fibonacci number

Phased Tilings and Generalized Fibonacci Identities

Phased Tilings and Generalized Fibonacci Identities

Random Approaches to Fibonacci Identities

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsArthur T. BenjaminARTHUR T. BENJAMIN is Associate Professor of Mathematics at Harvey Mudd College in Claremont, California, where he has taught since 1989 after earning his Ph.D. from Johns Hopkins in Mathematical Sciences. His research interests are in combinatorics and game theory. He recently received the 2000 Deborah and Franklin Tepper Haimo Award for Distinguished Teaching from the MAA.Gregory M. LevinGREGORY M. LEVIN received a B.S. in mathematics from Harvey Mudd College in 1992 and a Ph.D. in mathematical sciences from The Johns Hopkins University in 1997. He has spent the last two years as a visiting professor at Harvey Mudd. His mathematical interests are in discrete mathematics, and the majority of his research has been in fractional graph theory.Karl MahlburgKARL MAHLBURG is studying for a B.S. in mathematics at Harvey Mudd College, with expected graduation in 2001. His research interests include algebra, combinatorics, and topology, which he hopes to study more thoroughly at the graduate level.Jennifer J. QuinnJENNIFER J. QUINN is Associate Professor of Mathematics at Occidental College in Los Angeles. She earned her B.A., M.S., and Ph.D. from Williams College, University of Illinois, University of Chicago, and University of Wisconsin.

Fibonacci With a Golden Ring

(Fo = O, F1 = 1, 1, 2,3,5,8,13,..., Fn+2 =IFi + 1 + F..) in terms of the golden ratio 0 = (1 + V5)/2 and its conjugate (k = (1 V5)/2 (e.g., [4, p. 252]). We would like to show that if you play with the matrix A a little, adding, subtracting, multiplying, and exponentiating, you will soon find yourself in the higher domain 74 A] and rewarded with a spectacular view of much of the beautiful Fibonacci landscape, instead of just one formula. From there, you will be able to see a unified proof of a number of familiar Fibonacci identities, such as

A note on the improper use of a formula for Fibonacci numbers

The improper use of a formula for Fibonacci numbers gives rise to an interesting class of integers (namely, the numbers Gn (k))governed by the integral parameters nand k.After establishing some properties of these numbers, we extend them to negative values of the subscript nand use this extension to obtain a Fibonacci identity. A glimpse of a possible further generalization is also caught.

Schur Functions and Fibonacci Identities

Schur Functions and Fibonacci Identities

A Proof from Graph Theory for a Fibonacci Identity

A Proof from Graph Theory for a Fibonacci Identity

Associated Hyperbolic and Fibonacci Identities

Associated Hyperbolic and Fibonacci Identities

General Identities for Linear Fibonacci and Lucas Summations

General Identities for Linear Fibonacci and Lucas Summations

Exponential Generating Functions for Fibonacci Identities

Exponential Generating Functions for Fibonacci Identities

Generating Identities for Fibonacci and Lucas Triples

Generating Identities for Fibonacci and Lucas Triples

Some Generalized Fibonacci Identities

Some Generalized Fibonacci Identities