Two Implementations of Real-Time Sequence Generator for {n^3 | n=1, 2, 3, ... } and Their Comparison

A cellular automaton (CA) is a well-studied non-linear computational model of complex systems in which an infinite one-dimensional array of finite state machines (cells) updates itself in a synchronous manner according to a uniform local rule. A sequence generation problem on the CA model has been studied for a long time and a lot of generation algorithms has been proposed for a variety of non-regular sequences such as {2^n | n = 1, 2, 3,...}, prime, and Fibonacci sequences etc. In this paper, we study a real-time sequence generator for {n^3  | n=1, 2, 3, ...}. In the previous studies, Kamikawa and Umeo(2018) showed that sequence {n^3 | n=1, 2, 3, ...} can be generated in real-time by an eight-state CA. We present a new six-state implementation of real-time sequence generator for {n^3 | n=1, 2, 3, ...} rather than reducing the internal state of the Kamikawa and Umeo's sequence generator and give a formal proof of the correctness of the generator. In addition, we show the number of state-changes and number of cells of sequence generators, and compare sequence generators.