斐波那契数与三价图

斐波那契数与恒等式

Fibonacci Numbers and Identities

通过研究函数的一个递归关系，给出了关于斐波那契数和Lucas数的各种恒等式的替代证明，并通过与递归关系相关联的三价图，使某些著名的恒等式变得可见.

一类图的邻接矩阵与斐波那契数

The adjacency matrix of one type of graph and the Fibonacci numbers

近年来，图论引起了人们极大的兴趣，矩阵的整数幂的计算也得到了广泛的研究。本文研究了一类图与著名的Fibonacci数列之间的关系。本文考虑一类2k（k= 1，2，.）图的邻接矩阵.顶点。我们还知道，对于任何正整数r，A^{r}（A是图的邻接矩阵）的第（i，j）项就是从顶点i到顶点j的遍历次数，这些遍历恰好使用k条边。

Construction of helices from Lucas and Fibonacci sequences

By means of two complex-valued functions (depending on an integer parameter P>=1) we construct helices of integer ratio R>=1 related to the so-called Binet formulae for P-Lucas and P-Fibonacci sequences. Based on these functions a new map is defined and we show that its three-dimensional representation is also a helix. After proving that the lattice points of these later helix satisfy certain diophantine Pell's equations we call it a Pell's helix. We prove that for P-Fibonacci and Pell's helices the respective ratio is an invariant, contrasting to the P-Lucas helices whose ratio depends on P. …

Fibonacci-like sequences for variants of the tower of Hanoi, and corresponding graphs and gray codes

We modify the rules of the classical Tower of Hanoi puzzle in a quite natural way to get the Fibonacci sequence involved in the optimal algorithm of resolution, and show some nice properties of such a variant. In particular, we deduce from this Tower of Hanoi-Fibonacci a Gray-like code on the set of binary words without the factor 11, which has some properties intersting for itself and from which an iterative algorithm for the Tower of Hanoi-Fibonacci is obtained. Such an algorithm involves the Fibonacci substitution. Eventually, we briefly extend the study to some natural generalizations.

利用斐波那契数列和切比雪夫多项式表示轮图图表的Fox着色群与Alexander-Burau-Fox模

Using Fibonacci Numbers and Chebyshev Polynomials to Express Fox Coloring Groups and Alexander-Burau-Fox Modules of Diagrams of Wheel Graphs

本文利用斐波那契数列和切比雪夫多项式，计算轮图图表（即辫子闭包）的约化Fox着色群及Alexander-Burau-Fox模，推广了相关结果。

通过模图结构的Collatz轨迹的斐波那契定理

A Fibonacci theorem for Collatz trajectories via modular graph structure

本文通过模6的Collatz变换有向图G及其子图G'，证明在{1,...,2^m}中恰好有F(m+1)个奇数在T的轨道中第2至m步避免模6余4类，并构造了整数与有向路径的双射Ψ_m。