Cellular automata with accumulative memory:: Legal rules starting from a single site seed

Standard Cellular Automata (CA) are ahistoric (memoryless), i.e., the new state of a cell depends only on the neighborhood configuration at the preceding time step. This article introduces an extension of the standard framework of CA by considering automata implementing memory capabilities. While the update rules of the CA remains the same, each site remembers a weighted mean of all its past states, with a decreasing weight of states farther back in the past. The historic weighting is defined by a potential series of coefficients, t(k), k acting as a forgetting factor. This paper considers the time evolution of one-dimensional, legal CA rules with accumulative memory.