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PreprintCC BY 4.0arXiv

Asymptotics for partitions over the Fibonacci numbers and related sequences, II

  • Michael Coons
  • Simon Kristensen
  • Mathias L. Laursen
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Abstract

In a recent paper of the same title, we considered nondistinct partitions of integers over the Fibonacci sequence and related linearly recurrent sequences. In this paper, continuing this story, we apply the same method, but consider distinct partitions. In particular, let qF(m) denote the number of ways of writing m as the sum of distinct Fibonacci numbers. It is well-known that, while qF(m) behaves wildly, its partial sums behave in a more structured way. In particular, Kempton showed that the function N−log 2/logφ∑mleqslant NqF(m) is asymptotically logarithmically-periodic. In this paper, we improve on this result by providing an explicit error term. Additionally, we prove the analogous result for a class of more general linear recurrent sequences.

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Cite

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BibTeX
@misc{coons2026asymptotics,
  title = {{Asymptotics for partitions over the Fibonacci numbers and related sequences, II}},
  author = {Michael Coons and Simon Kristensen and Mathias L. Laursen},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04869},
  eprint = {2611.04869},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04869},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Asymptotics for partitions over the Fibonacci numbers and related sequences, II
AU  - Michael Coons
AU  - Simon Kristensen
AU  - Mathias L. Laursen
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04869
UR  - https://arxiv.org/abs/2611.04869
AB  - In a recent paper of the same title, we considered nondistinct partitions of integers over the Fibonacci sequence and related linearly recurrent sequences. In this paper, continuing this story, we apply the same method, but consider distinct partitions. In particular, let q_F(m) denote the number of ways of writing m as the sum of distinct Fibonacci numbers. It is well-known that, while q_F(m) behaves wildly, its partial sums behave in a more structured way. In particular, Kempton showed that the function N^−log 2/logφ∑_mleqslant Nq_F(m) is asymptotically logarithmically-periodic. In this paper, we improve on this result by providing an explicit error term. Additionally, we prove the analogous result for a class of more general linear recurrent sequences.
ER  -