Non-Fisherian generalized Fibonacci numbers

Thor Martinsen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 370–389
DOI: 10.7546/nntdm.2025.31.2.370-389
Full paper (PDF, 283 Kb)

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Authors and affiliations

Thor Martinsen
Department of Applied Mathematics, Naval Postgraduate School
1 University Circle, Monterey, California, 93943, USA

Abstract

Using biology as inspiration, this paper explores a generalization of the Fibonacci sequence that involves gender biased sexual reproduction. The female, male, and total population numbers along with their associated recurrence relations are considered. We demonstrate that the generalized Fibonacci numbers being investigated are generalized third order Pell–Lucas numbers. Sequence properties, generating functions, and closed-form solutions for these new generalized Fibonacci numbers, as well as several identities involving Jacobsthal, Leonardo, and generalized Leonardo numbers are presented. The generalized Fibonacci number framework developed gives rise to many previously uncataloged sequences, and develops new connections between known sequences.

Keywords

• Generalized Fibonacci numbers
• Generalized third order Pell–Lucas numbers
• Jacobsthal numbers
• Leonardo numbers
• Fisher’s principle
• Extraordinary sex ratios

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 92D25

References

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Manuscript history

• • Received: 30 November 2024
  • Revised: 4 June 2025
  • Accepted: 9 June 2025
  • Online First: 14 June 2025
  • Correction Notice: 31 March 2026
    Per Author’s request made on 31 March 2026, the following corrections are made:
    on page 373, line -11: Old:   New:
    on page 388, line 9: Old: 577   New: 574
    on page 388, line -17: Old: 202  New: 212
    on page 388, line -13: Old: 3754  New: 4094
    on page 388, line -6: Old: 60, …, 5496  New: 53, …, 8213
    on page 388, line -5: Old: 100, …, 11656  New: 93, …, 14373
    on page 388, line -2: Old: 4971, 10766 New: 4471, 10266
    on page 388, line -1: Old: 7654, 16926  New: 7154, 16426
    on page 389, line 9: Old: 41  New: 42

Copyright information

Ⓒ 2025 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).