In case (b), with rows chosen to be 2 j elements in length, the leftmost column will always be identical to the beginning of the sequence, and in addition every interior element will be black exactly when the cell at the top of its column has the same color as the one at the beginning of its row.

These numbers can also be obtained as the coefficients of x n in the series expansion of x ∂ x Log[ ζ [m, x]] , with the so-called zeta function, which is always a rational function of x , given by ζ [m_, x_] := 1/Det[IdentityMatrix[Length[m]] - m x] and corresponds to the product over all cycles of 1/(1 - x n ) .

Implementation [of causal networks] Given a list of successive positions of the active cell, as from Map[Last, MAEvolveList[rule, init, t]] (see page 887 ), the network can be generated using MAToNet[list_] := Module[{u, j, k}, u[_] = ∞ ; Reverse[ Table[j = list 〚 i 〛 ; k = {u[j - 1], u[j], u[j + 1]}; u[j - 1] = u[j] = u[j + 1] = i; i  k, {i, Length[list], 1, -1}]]] where nodes not yet found by explicit evolution are indicated by ∞ .

It implies that every block of length m that occurs at a particular step has exactly 4 immediate predecessor blocks of length m + 2 (see page 960 ). … The length of the longest cycle grows roughly like 2 0.63 n (see page 260 ).

Hump m in the picture of sequence (c) shown is given by FoldList[Plus, 0, Flatten[Nest[Delete[NestList[Rest, #, Length[#] - 1], 2]&, Append[Table[1, {m}], 0], m]] - 1/2] The first 2 m elements in the sequence can also be generated in terms of reordered base 2 digit sequences by FoldList[Plus, 1, Map[Last[Last[#]]&, Sort[Table[{Length[#], Apply[Plus, #], 1 - #}& [ IntegerDigits[i, 2]], {i, 2 m }]]]] Note that the positive and negative fluctuations in sequence (f) are not completely random: although the probability for individual fluctuations in each direction seems to be the same, the probability for two positive fluctuations in a row is smaller than for two negative fluctuations in a row. … The total lengths of these chains (corresponding to the depth of the evaluation tree) seem to increase roughly like Log[n] for all the rules on this page.

(The size of the region before stripes appear depends on Length[ContinuedFraction[θ/π]] .)

But given t steps in this sequence as a list of 0's and 1's, the following function will reconstruct the rightmost t digits in the starting value of n : IntegerDigits[First[Fold[{Mod[If[OddQ[#2], 2 First[#1] - 1, 2 First[#1] PowerMod[5, -1, Last[#1]]], Last[#1]], 2 Last[#1]} &, {0, 2}, Reverse[list]]], 2, Length[list]]

For large n , the average number of distinct cycles in all such networks is Sqrt[ π /2] Log[n] , and the average length of these cycles is Sqrt[ π /8 n] .

EvenQ] := Partition[ Fold[Insert[#1, #2, Random[Integer, Length[#1]] + 1] &, {}, Floor[Range[1, n + 2/3, 1/3]]], 2] Networks obtained in this way are usually connected, but will almost always contain self-loops and multiple edges.

Register machines [from cellular automata] Given the program for a register machine in the form used on page 896 , the rules for a cellular automaton that emulates it can be obtained from g[i[1], p_, m_] := {{_, p, _}  p + 1, {_, 0, p}  m + 2, {_, _, p}  m + 3} g[i[2], p_, m_] := {{_, p, _}  p + 1, {p, 0, _}  m + 5, {p, _, _}  m + 6} g[d[1, q_], p_, m_] := {{m + 2 | m + 3, p, _}  q, {m + 1, p, _}  p, {0, p, _}  p + 1, {_, m + 2 | m + 3, p}  m + 1} g[d[2, q_], p_, m_] := {{_, p, m + 5 | m + 6}  q, {_, p, m + 4}  p, {_, p, 0}  p + 1, {p, m + 5 | m + 6, _}  m + 4} RMToCA[prog_] := With[{m = Length[prog]}, Flatten[ {MapIndexed[g[#1, First[#2], m] &, prog], {{0, 0 | m + 1, m + 3}  m + 2, {0, m + 1, _}  0, {0, 0, m + 1}  0, {_, _, x : (m + 1 | m + 3)}  x, {_, m + 1 | m + 3, _}  m + 2, {m + 6, 0 | m + 4, 0}  m + 5, {_, m + 4, 0}  0, {m + 4, 0, 0}  0, {x : (m + 4 | m + 6), _, _}  x, {_, m + 4 | m + 6, _}  m + 5, {_, x_ , _}  x}}]] If m is the length of the register machine program, then the resulting cellular automaton has m + 7 possible colors for each cell.