细胞自动机中的牙签序列及其他序列
The Toothpick Sequence and Other Sequences from Cellular Automata
摘要：
牙签的二维排列通过以下迭代过程构造。在第一阶段，将一根长度为1的牙签放在一个正方形网格上，与y轴对齐。在随后的每个阶段，对于每个暴露的牙签末端，将一根正交牙签放在该末端的中心。所得到的结构具有类似分形的外观。我们将分析牙签序列，它给出了n步后牙签的总数。我们还研究了几个相关的序列，所产生的细胞自动机中的活动细胞计数。一些不寻常的复发出现：一个典型的例子是，不是斐波那契递归，我们可以写为a（2+i）= a（i）+ a（i+1），我们设置n = 2^k+i（0 <= i < 2^k），然后a（n）=a（2^k+i）=2a（i）+a（i+1）。相应的生成函数看起来像Prod{k >= 0}（1+x^{2^k-1}+2x ^^2^k}）及其变体。
英文摘要：
A two-dimensional arrangement of toothpicks is constructed by the following iterative procedure. At stage 1, place a single toothpick of length 1 on a square grid, aligned with the y-axis. At each subsequent stage, for every exposed toothpick end, place an orthogonal toothpick centered at that end. The resulting structure has a fractal-like appearance. We will analyze the toothpick sequence, which gives the total number of toothpicks after n steps. We also study several related sequences that arise from enumerating active cells in cellular automata. Some unusual recurrences appear: a typical example is that instead of the Fibonacci recurrence, which we may write as a(2+i) = a(i) + a(i+1), we set n = 2^k+i (0 <= i < 2^k), and then a(n)=a(2^k+i)=2a(i)+a(i+1). The corresponding generating functions look like Prod{k >= 0} (1+x^{2^k-1}+2x^{2^k}) and variations thereof.
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