Tiling circular $n$-boards with squares and (1,1)-fences

Michael A. Allen; Kenneth Edwards

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arXiv · 2610.39599

Tiling circular $n$-boards with squares and (1,1)-fences

Michael A. Allen·Kenneth Edwards

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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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Michael A. Allen, Kenneth Edwards. 2026-10-29. Tiling circular $n$-boards with squares and (1,1)-fences. https://arxiv.org/abs/2610.39599

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