StatisticalMechanics / CONCEPT

ON cells of the Rule 13 cellular automaton

The elementary cellular automaton with Wolfram rule 13, started from a single ON cell at the origin and updated at every cell of the integers, has n + 1 ON cells among the cells -2n, ..., 2n of row 2n and 3n + 1 ON cells among the cells -(2n + 1), ..., 2n + 1 of row 2n + 1, as conjectured by Ctibor O. Zizka for OEIS A266285; the counts therefore satisfy Colin Barker's closed form ((-1)^n (3 - 2n) + 4n + 1) / 4, his recurrence a(n) = 2 a(n - 2) - a(n - 4) for n > 3 and his generating function (1 + x + 2x^3) / ((1 - x)^2 (1 + x)^2).