PreprintCC BY 4.0arXiv
Tiling circular n-boards with squares and (1,1)-fences
Abstract
We show that the number of ways to tile a circular n-board (an annulus divided into n curved unit square cells) with appropriately curved unit squares and curved (1,1)-fences is a Lucas number squared if n is even and an odd-indexed Lucas number otherwise. A curved (1,1)-fence is composed of two curved unit squares separated by a gap in the shape of another curved unit square. This combinatorial interpretation of the Lucas numbers squared and the odd-indexed Lucas numbers allows us to obtain various identities concerning these sequences in a quick and lucid manner.
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BibTeX
@misc{allen2026tiling,
title = {{Tiling circular $n$-boards with squares and (1,1)-fences}},
author = {Michael A. Allen and Kenneth Edwards},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39599},
eprint = {2610.39599},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39599},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tiling circular n-boards with squares and (1,1)-fences
AU - Michael A. Allen
AU - Kenneth Edwards
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39599
UR - https://arxiv.org/abs/2610.39599
AB - We show that the number of ways to tile a circular n-board (an annulus divided into n curved unit square cells) with appropriately curved unit squares and curved (1,1)-fences is a Lucas number squared if n is even and an odd-indexed Lucas number otherwise. A curved (1,1)-fence is composed of two curved unit squares separated by a gap in the shape of another curved unit square. This combinatorial interpretation of the Lucas numbers squared and the odd-indexed Lucas numbers allows us to obtain various identities concerning these sequences in a quick and lucid manner.
ER -