A combinatorial study of the generalized Leonardo polynomials

Mark Shattuck
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 378–394
DOI: 10.7546/nntdm.2026.32.2.378-394
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Authors and affiliations

Mark Shattuck
Department of Mathematics, University of Tennessee
Knoxville, TN 37996, USA

Abstract

Let \mathcal{GL}_{n}(x)=\mathcal{GL}_{n}^{a,b,T}(x) denote the sequence satisfying the recursion \mathcal{GL}_{n}(x)=x\mathcal{GL}_{n-1}(x)+\mathcal{GL}_{n-2}(x)+T for n \geq 2, with \mathcal{GL}_{0}(x)=a and \mathcal{GL}_{1}(x)=bx, where a, b and T are arbitrary. The \mathcal{GL}_n(x) are referred to as generalized Leonardo polynomials. In this paper, we provide a combinatorial interpretation for \mathcal{GL}_{n}(x) as a weight function on a certain class of linear tilings wherein the initial tiles are colored according to their type. We use our interpretation to provide combinatorial explanations of some identities for \mathcal{GL}_{n}(x) which were previously found by various algebraic methods. Further, we are able to establish by combinatorial arguments several new identities for \mathcal{GL}_{n}(x), among them, formulas for various sums of products and a Cassini-like relation.

Keywords

  • Leonardo number
  • Leonardo polynomial
  • Linear tiling
  • Combinatorial proof

2020 Mathematics Subject Classification

  • 05A19
  • 11B39

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Manuscript history

  • Received: 24 April 2026
  • Revised: 24 August 2026
  • Accepted: 26 August 2026
  • Online First: 27 August 2026

Copyright information

Ⓒ 2026 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).

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Cite this paper

Shattuck, M. (2026). A combinatorial study of the generalized Leonardo polynomials. Notes on Number Theory and Discrete Mathematics, 32(2), 378-394, DOI: 10.7546/nntdm.2026.32.2.378-394.

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