On the evolution of the cellular automaton of rule 150 from some simple initial states
The evolution Ea in discrete time t of the cellular automaton CA150 of Wolfram rule 150 is studied for certain initial states psa. The values of the cell variables in any state psa considered are all zero outside a principal sequence (PS) of length k within which they take on values 1a2a3ak 2ak 11, where a2=1,0, Powerful algebraic methods are used to develop formulas for the weight, Oa(t) i.e. the total number of cell variables with value equal to 1 at time t, of the state of Ea at time t>=0, and of the sum Wa(N)=t=02N 1Oa(t) of the weights of the first 2N states of the evolution of Ea in discrete time. The methods are described carefully. A selection of results, new here, is stated and proved for several initial states psa with simple PSs of increasing lengths k>1. Many of them feature numbers in the Fibonacci, Lucas, and Jacobsthal sequences.