Real-time Generation of Polygonal Sequences by One-Dimensional Cellular Automata
Alexis García Durán and José Manuel Gómez Soto
Let {𝑃𝑠𝑛}∞𝑛=1 be the sequence of polygonal numbers where 𝑃𝑠𝑛 = (𝑠 − 2) 𝑛(𝑛−1)/2 + 𝑛 with 𝑠 ≥ 3 is an integer. In this paper, we describe two wave networks in cellular automata that can generate the sequences {𝑃𝑠𝑛}∞𝑛=1 in real-time. The first is a wave network that yields a cellular automaton with 𝑠 2 + 2 states that generates this sequence when 𝑠 ≥ 4 is an even integer. With the second wave network, we obtain a cellular automaton with 𝑠 states when 𝑠 ≥ 5 is an odd integer, or a cellular automaton with four states when 𝑠 = 3.
Keywords: Cellular automata, real-time sequence generation problem
DOI: 10.32908/jca.v18.020224