A combinatorial approach for constructing non-monotonic solutions to the generalized Golomb recursion

Sunohara, Matthew; Tanny, Stephen M.; Verberne, Yvon

doi:10.1080/10236198.2019.1582648

Cited by 2 publications

(2 citation statements)

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“…Finally, there are other sequences besides those related to Conolly's sequence that satisfy nested recurrences and are connected to trees [10,13,14,23], such as Golomb's sequence G…”

Section: Future Workmentioning

confidence: 99%

“…These trees are constructed recursively; we explain the construction and labeling process in Section 1.1. Classically, sequences arising from trees are necessarily slow, though some recent variations on the treebased methodology have allowed generating non-slow sequences from trees as well [14,23]. Such combinatorial interpretations often lead directly to an asymptotic description of the sequence.…”

Section: Introductionmentioning

confidence: 99%

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Connecting Slow Solutions to Nested Recurrences with Linear Recurrent Sequences

Fox

2022

Preprint

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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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“…Finally, there are other sequences besides those related to Conolly's sequence that satisfy nested recurrences and are connected to trees [10,13,14,23], such as Golomb's sequence G…”

Section: Future Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%