One-dimensional game of life and its growth functions

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={1 ￠ € ‰ ￠ € ‰ ￠ € ￠ € ￠ € for ￠ € ‰ ￠ € ‰ ￠ € ‰j=0,k ￠ € ￠ € ￠ € ￠ € ￠ € ￠ € ￠ € ￠ € ￠ € (I)0 ￠ € ‰ ￠ € ‰ ￠ € ‰or ￠ € ‰ ￠ € ‰ ￠ € ‰1 ￠ € ‰ ￠ € ‰ ￠ € ‰for ￠ € ‰ ￠ € ‰ ￠ € ‰0k ￠ € ￠ € ￠ € ￠ € ￠ € ￠ € ￠ € ￠ € ‰ ￠ € ‰ ￠ € ‰ ￠ € ‰ ￠ € ‰ ￠ € ￠ € (II)di+1,j=di,j+1(mod2) ￠ € ‰ ￠ € ‰ ￠ € ‰for ￠ € ‰ ￠ € ‰ ￠ € ‰i ￠ ‰ ￥0. ￠ € ￠ € ￠ € ￠ € ￠ € ￠ € (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=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.