用线性递归序列连接嵌套递归的慢解
Connecting Slow Solutions to Nested Recurrences with Linear Recurrent Sequences
摘要：
标号无限树为嵌套递归关系生成的许多整数序列提供了组合解释。通常，这样的序列是单调递增的。这些序列中有几个还对序列中每个值出现的频率进行了简单的描述。在本文中，我们推广了最经典的例子，一个更大的家庭的序列参数化的线性递归关系。我们的每一个序列都可以用三种不同的方式构造：通过嵌套递归关系，从标记的无限树，或者通过使用Zeckendorf样的数字串来描述它的频率序列。最后讨论了序列的渐近性质。
英文摘要：
Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward descriptions in terms of how often each value in the sequence occurs. In this paper, we generalize the most classical examples to a larger family of sequences parametrized by linear recurrence relations. Each of our sequences can be constructed in three different ways: via a nested recurrence relation, from labeled infinite trees, or by using Zeckendorf-like strings of digits to describe its frequency sequence. We conclude the paper by discussing the asymptotic behaviors of our sequences.
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