论文标题
Zeckendorf的定理使用算术进展中的索引
Zeckendorf's Theorem Using Indices in an Arithmetic Progression
论文作者
论文摘要
Zeckendorf的定理指出，任何正整数都可以独特地分解为不同的，非阳性的斐波那契数。有许多概括，包括仅使用索引的斐波那契数的分解的结果。我们进一步扩展了这些，并证明仅在给定算术进程中使用指标时就会产生相似的结果。作为我们证明的一部分，我们为斐波那契数字生成了一系列新的复发，这些数字本身就是感兴趣的。
Zeckendorf's Theorem states that any positive integer can be uniquely decomposed into a sum of distinct, non-adjacent Fibonacci numbers. There are many generalizations, including results on existence of decompositions using only even indexed Fibonacci numbers. We extend these further and prove that similar results hold when only using indices in a given arithmetic progression. As part of our proofs, we generate a range of new recurrences for the Fibonacci numbers that are of interest in their own right.
