Fibonacci数列及其推广形式的正整数表示

Representation of Natural Numbers Using Generalized Fibonacci Sequence

摘要: 正整数表示问题前人多有研究，而基于Fibonacci数列及其推广形式的分析并不多见。本文的主要工作是探讨了该类整数表示的可行性，发现了表示的多样性，从而从最少表示及最多表示的角度来展开分析，分别引入了它们的计数多项式以及0与1的编码形式。起初是只针对Fibonacci数列，之后研究Lucas数列的情况，再接着对一类推广：n代的Fibonacci数列做了猜测。

Abstract: Previously, there were many studies about the problem of representation of natural numbers. But it’s comparatively rare to study the problem based on Fibonacci sequence and its extension. This thesis mainly discussed the feasibility and diversity of this kind of representation. Utilizing enu-merating polynomials and binomial codes, we focused on minimal and maximal representations of natural numbers. In addition to Fibonacci sequence, we also studied the situation of Lucas sequence and offered some hypotheses in the case of n-step Fibonacci sequence.