International Journal of Mathematics and Mathematical SciencesVolume 15, Issue 3, Pages 499-508http://dx.doi.org/10.1155/S0161171292000656

One-dimensional game of life and its growth functions

Department of Mathematics and Computer Science, University cf Wisconsin, Whitewater 53190, WI, USA

Received 6 June 1990; Revised 28 October 1991

Copyright © 1992 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 d1,j recursively for 1, a non-negative integer, and j an arbitrary integer by the rules:d0,j={1forj=0,k(I)0or1for0<j<kd0,j=0forj<0orj>k(II)di+1,j=di,j+1(mod2)fori0.(III)Now, if we interpret the number of 1's in row i as the coefficient ai of a formal power series, then we obtain a growth function, f(x)=i=0aixi. 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.