Mohammad H. Ahmadi, "One-dimensional game of life and its growth functions", International Journal of Mathematics and Mathematical Sciences, vol. 15, Article ID 525863, 10 pages, 1992. https://doi.org/10.1155/S0161171292000656
One-dimensional game of life and its growth functions
We start with finitely many 's and possibly some 's in between. Then each entry in the other rows is obtained from the Base sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Define recursively for , a non-negative integer, and an arbitrary integer by the rules:Now, if we interpret the number of 's in row as the coefficient of a formal power series, then we obtain a growth function, . It is interesting that there are cases for which this growth function factors into an infinite product of polynomials. Furthermore, we shall show that this power series never represents a rational function.
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