One-dimensional game of life and its growth functions
We start with finitely many1's and possibly some0's in between. Then each entry in the other rows is obtained from the Base2sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Defined1,jrecursively for1, a non-negative integer, andjan arbitrary integer by the rules:d0,j={1 for j=0,k (I)0 or 1 for 0<j<kd0,j=0 for j<0 or j>k (II)di+1,j=di,j+1(mod2) for i≥0. (III)Now, if we interpret the number of1's in rowias the coefficientaiof a formal power series, then we obtain a growth function,f(x)=∑i=0∞aixi. 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.