Fibonacci三角形

利用 Pell方程和递推序列的方法证明了在 k=1 ,2 ,3 ,4,5时 ,以 Fibonacci数 Fn,Fn,Fn- k为边的Fibonacci三角形不存在 .     

Fibonacci integers

A Fibonacci integer is an integer in the multiplicative group generated by the Fibonacci numbers. For example, 77=21⋅55/(3⋅5) is a Fibonacci integer. Using some results about the structure of this multiplicative group, we determine a near-asymptotic formula for the counting function of the Fibonacci integers, showing that up to x the number of them is between and , for an explicitly determined constant c. The proof is based on both combinatorial and analytic arguments.     

Fibonacci multiplication

A curious binary operation on the nonnegative integers is shown to be associative.     

Fibonacci cubes

Two visual proofs illustrating a cubic Fibonacci number identity are given. These provide a basis for further activities and consideration of the use of visual approaches to other identities.     

Fibonacci代数的Hochschild同调群

设Λd是Fibonacci代数,基于对Bardzell极小投射双模分解的细致分析,用组合的方法清晰地计算了Fibonacci代数Λd的各阶Hochschild同调群的维数.     

Fibonacci solitaire

Fibonacci Solitaire is a combinatorial algorithm which associates with a permutation of [n]={1,…,n} a partition of [n] into couples and singletons. We study the output configuration when the algorithm is applied to a random permutation, with emphasis on the large n‐asymptotics. We show that the set of singletons, properly scaled, resembles a familiar ‘stick‐breaking’ Poisson configuration, whereas the configuration of couples becomes close to uniform. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 71–88, 2002     

The Fibonacci length of certain centro-polyhedral groups

We examine the Fibonacci length of certain centro-polyhedral groups and show that in some cases the lengths depend on tribonacci sequences. Further we obtain specific examples of infinite families of threegenerator groups with constant, linear and (3-step) Wall number dependent Fibonacci lengths.     

Generalised Fibonacci manifolds

Fibonacci manifolds have a hyperbolic structure which may be defined via Fibonacci numbers. Using related sequences of Lucas numbers, other 3-manifolds are constructed, their geometric structures determined, and a curious relationship between the homology and the invariant trace-field examined.Supported by the Royal Society.     

Split Fibonacci Quaternions

Starting from ideas given by Horadam in [5] , in this paper, we will define the split Fibonacci quaternion, the split Lucas quaternion and the split generalized Fibonacci quaternion. We used the well-known identities related to the Fibonacci and Lucas numbers to obtain the relations between the split Fibonacci, split Lucas and the split generalized Fibonacci quaternions. Moreover, we give Binet formulas and Cassini identities for these quaternions.     

A Fibonacci quotient

In this article, we describe the outcome of a mathematical collaboration between a university lecturer and an undergraduate student. The resulting investigation concerned a particular divisibility property of the Fibonacci numbers, and indeed it seems that a new result was found in this regard. An interesting point to be made here is that, although the mathematical content was relatively straightforward, this joint exploration did, in a very modest sense, mirror certain key aspects of the research process.     

广义Fibonacci数的一些求和公式

This note provides the some sum formulas for generalized Fibonacci numbers. The results are proved using clever rearrangements, rather than using induction.     

Fibonacci lattice points

Odd indexed Fibonacci numbers F2n+1 can be written as sums of two squares a ²+b ². In this paper, we study the distribution of the lattice points (a,b) on the circles of radius \(\sqrt{F_{2n}+1}\).