One-dimensional game of life and its growth functions

Mohammad H. Ahmadi

International Journal of Mathematics and Mathematical Sciences, 1992, vol. 15, 1-10

Abstract:

We start with finitely many 1 's and possibly some 0 's in between. Then each entry in the other rows is obtained from the Base 2 sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Define d 1 , j recursively for 1 , a non-negative integer, and j an arbitrary integer by the rules: d 0 , j = { 1 for j = 0 , k ( I ) 0 or 1 for 0 < j < k d 0 , j = 0 for j < 0 or j > k ( I I ) d i + 1 , j = d i , j + 1 ( mod 2 ) for i ≥ 0. ( I I I ) Now, if we interpret the number of 1 's in row i as the coefficient a i of a formal power series, then we obtain a growth function, f ( x ) = ∑ i = 0 ∞ a i x i . 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.

Date: 1992
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:525863

DOI: 10.1155/S0161171292000656

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