AI 中文总结
该研究给出三行正方形格条上自回避多边形的计数闭式公式，推导相关子类型的二项式和表示，通过两种表示得到恒等式的新几何证明，并建立其与序列A007909的关联。
AI 中文摘要
我们给出了格条$S_2:=\mathbb{Z}\times\{0,1,2\}$上长度为$n$的自回避多边形（SAP）的数量$p^{S_2}(n)$的闭式公式，以及由最左列和最右列的竖步数确定的SAP子类型的闭式公式。对于最左列和最右列各含两个竖步的子类型，我们还推导了其作为二项式和的另一种表示形式。我们的推导是初等的：完全是组合和几何的，避免了生成函数。比较这两种表示形式，得到了对Larsen研究中出现的一个恒等式的新几何证明，该恒等式源于Gessel提出的问题。最后，我们证明该子类型的SAP与序列A007909密切相关。更准确地说，对于$m\geq0$，最左列和最右列各含两个竖步且长度为$2m+6$的这类SAP的数量，由该序列索引为$m$的项给出，从而该序列除了其枚举的合成数外，还获得了几何解释。
英文摘要
We give a closed formula for the number $p^{S_2}(n)$ of self-avoiding polygons (SAPs) of length $n$ on the strip $S_2:=\mathbb{Z}\times\{0,1,2\}$, together with closed formulas for those subtypes of SAPs which are determined by the numbers of vertical steps in their leftmost and rightmost columns. For the subtype whose leftmost and rightmost columns each contain two vertical steps, we also derive an alternative representation as a binomial sum. Our derivation is elementary: it is purely combinatorial and geometric and avoids generating functions. Comparing the two representations yields a new geometric proof of an identity arising in Larsen's treatment \cite{L07} of a problem posed by Gessel \cite{G95}. Finally, we show that this subtype of SAPs is closely connected to the sequence A007909. More precisely, for $m\geq0$, the number of these SAPs whose leftmost and rightmost columns each contain two vertical steps and whose length equals $2m+6$ is given by the term of this sequence with index $m$, which thereby acquires a geometric interpretation alongside the compositions it enumerates.
