Most of the 2 n possible states have unique predecessors; for large n , about 2 0.76 n or Root[# 3 - # 2 - 2 &, 1] n instead have 0 or 2 predecessors. The predecessors of a given state can be found from Cases[Map[Fold[Prepend[#1, If[#2  1 ⊻ , Take[#1, 2]  {0, 0}], 0, 1]] &, #, Reverse[list]] &, {{0, 0}, {0, 1}, {1, 0}, {1, 1}}], {a_, b_, c___, a_, b_}  {b, c, a}]

Some integer functions can readily be obtained by supplying integer arguments to continuous functions, so that for example Mod[x, 2] corresponds to Sin[ π x/2] 2 or (1 - Cos[ π x])/2, Mod[x, 3] ↔ 1 + 2/3(Cos[2/3 π (x - 2)] - Cos[2 π x/3]) Mod[x, 4] ↔ (3 - 2 Cos[ π x/2] - Cos[ π x] - 2 Sin[ π x/2])/2 Mod[x, n] ↔ Sum[j Product[(Sin[ π (x - i - j)/n]/ Sin[ π i/n]) 2 , {i, n - 1}], {j, n - 1}] (As another example, If[x > 0, 1, 0] corresponds to 1 - 1/Gamma[1 - x] .) And in this way the discrete system x  If[EvenQ[x], 3x/2, 3(x + 1)/2] from page 122 can be emulated by the continuous iterated map x  (3 + 6 x - 3 Cos[ π x])/4 .

At each step it compares the values of r and s , and if r is larger than s it replaces r and s by 4r – 4s – 1 and 2s + 1 respectively; otherwise it replaces them just by 4r and 2s .

The number of distinct sequences at step t in these three systems is respectively Ceiling[t/2] , t and Fibonacci[t+1] (which increases approximately like 1.618 t ).

Properties of [initially random cellular automaton] patterns For a random initial condition, the average density of black cells is exactly 1/2. For rule 126, the density after many steps is still 1/2. … For rule 30 and rule 150 it is exactly 1/2, while for rule 182 it is 3/4.

Corresponding to the result on page 870 for rule 90, the number of black cells at row t in the pattern from rule 150 is given by Apply[Times, Map[(2 # + 2 - (-1) # + 2 )/3 &, Cases[Split[IntegerDigits[t, 2]], k:{1 ..}  Length[k]]]] There are a total of 2 m Fibonacci[m+2] black cells in the pattern obtained up to step 2 m , implying fractal dimension Log[2, 1 + Sqrt[5]] . … The cell at position n on row t turns out to be given by Mod[GegenbauerC[n, -t, -1/2], 2] , as discussed on page 612 .

Labelling each shape and orientation with a different color, the behavior of this system can be reproduced with equal-sized squares using the rule {3  {{1, 0}, {3, 2}}, 2  {{1}, {3}}, 1  {{3, 2}}, 0  {{3}}} starting from initial condition {{3}} .

The average fraction of nodes that have no predecessor is (1 - 1/n) n or 1/ in the limit n  ∞ .

. • (b) All strings of length n containing exactly one black cell are produced—after at most 2n - 1 steps. • (c) All strings containing even-length runs of white cells are produced. • (d) The set of strings produced is complicated. … Strings of length n take n steps to produce. • (g) The same strings as in (f) are produced, but now a string of length n with m black elements takes n + m - 1 steps. • (h) All strings appear in which the first run of black elements is of length 1; a string of length n with m black elements appears after n + m - 1 steps. • (i) All strings containing an odd number of black elements are produced; a string of length n with m black cells occurs at step n + m - 1 . • (j) All strings that end with a black element are produced. • (k) Above length 1, the strings produced are exactly those starting with a white element. Those of length n appear after at most 3n - 3 steps. • (l) The same strings as in (k) are produced, taking now at most 2n + 1 steps. • (m) All strings beginning with a black element are produced, after at most 3n + 1 steps. • (n) The set of strings produced is complicated, and seems to include many but not all that do not end with . • (o) All strings that do not end in are produced. • (p) All strings are produced, except ones in which every element after the first is white.

Square root of rule 30 Although rule 30 cannot apparently be decomposed into other k = 2 , r = 1 cellular automata, it can be viewed as the square of the k = 3 , r = 1/2 cellular automata with rule numbers 11736, 11739 and 11742.