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PreprintCC BY 4.0arXiv

Generalized Zeckendorf expansions of the 2nd order

  • Sungkon Chang
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Abstract

We generalize Zeckendorf's theorem to second-order linear recurrences of the form Gk+2 = gGk+1 + hGk with arbitrary, coprime initial values (G1, G2). We analyze the asymptotic behavior of the count #RG(X) of positive integers less than or equal to X that have an expansion under the standard rule of expansions ℰ associated with the recurrence. Additionally, we provide sufficient conditions under which G has the unique expansion property under the rule of expansions. Finally, we completely characterize the g-golden ratio recurrence family when G1 = 1 or G2 = 1. Our approach directly investigates the algebraic structure of the expansions, simplifying and completing previous partial results.

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Cite

BibTeXRIS
BibTeX
@misc{chang2026generalized,
  title = {{Generalized Zeckendorf expansions of the 2nd order}},
  author = {Sungkon Chang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06795},
  eprint = {2609.06795},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06795},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Generalized Zeckendorf expansions of the 2nd order
AU  - Sungkon Chang
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06795
UR  - https://arxiv.org/abs/2609.06795
AB  - We generalize Zeckendorf's theorem to second-order linear recurrences of the form G_k+2 = gG_k+1 + hG_k with arbitrary, coprime initial values (G_1, G_2). We analyze the asymptotic behavior of the count #R_G(X) of positive integers less than or equal to X that have an expansion under the standard rule of expansions ℰ associated with the recurrence. Additionally, we provide sufficient conditions under which G has the unique expansion property under the rule of expansions. Finally, we completely characterize the g-golden ratio recurrence family when G_1 = 1 or G_2 = 1. Our approach directly investigates the algebraic structure of the expansions, simplifying and completing previous partial results.
ER  -