AI 中文总结
本文推导经典桥与火炬问题及容量为3的桥的最优过桥时间闭式表达式，得到新整数序列A392834，还针对星型图探索该问题并恢复相关经典恒等式。
AI 中文摘要
针对具有过桥时间集合{1,…,n}的经典桥与火炬问题，我们利用该问题最优子结构推导的递推关系，得到最优过桥时间T(n)的闭式表达式，该式对所有n≥2成立。我们将该问题推广到容量为3的桥，得到最优过桥时间T₃(n)，该式对所有n≥7成立。由此，我们在整数序列在线百科全书中获得新序列A392834。最后，我们还针对星型图探索该问题，并展示如何恢复一些涉及向下取整函数和的经典恒等式。
英文摘要
For the classic bridge and torch problem with crossing times $\left\{1,\ldots,n\right\}$, we derive a closed-form expression for the optimal crossing time $T\left(n\right)$ using a recurrence relation derived from the problem's optimal substructure, thus obtaining \[T\left(n\right)=\frac{n^2}{4}+3n-5+\frac{\left(-1\right)^n-1}{8}\] which holds for all $n\ge 2$. We generalize the problem to a bridge of capacity 3 and obtain the optimal crossing time \[T_3\left(n\right)=\frac{n^{2}}{6}+2n-\frac{181}{36}+\frac{\left(-1\right)^{n}}{4}-\frac{2}{9}\cos\left(\frac{2nπ}{3}\right)\] which holds for all $n\ge 7$. As such, we obtain a new sequence A392834 in the On-Line Encyclopedia of Integer Sequences. Lastly, we also explore this problem for star graphs, and see how we can recover some classic identities involving the sum of floor functions.
