LFSR cryptanalysis Given a sequence obtained from a length n LFSR (see page 975 ) Nest[Mod[Append[#, Take[#, -n] . vec], 2] &, list, t] the vector of taps vec can be deduced from LinearSolve[Table[Take[seq, {i, i + n - 1}], {i, n}], Take[seq, {n + 1, 2n}], Modulus  2] (An iterative algorithm in n taking about n 2 rather than n 3 steps was given by Elwyn Berlekamp and James Massey in 1968.)

The quantity FoldList[Plus, 0, Table[MoebiusMu[i], {i, n}]] behaves very much like a random walk.

Multiway systems based on numbers One can consider for example the rule n  {n + 1, 2 n} implemented by NestList[Union[Flatten[{# + 1, 2 #}]] &, {0}, t] In this case there are Fibonacci[t + 2] distinct numbers obtained at step t .

3D class 4 [cellular automaton] rules With a cubic lattice of the type shown on page 183 , and with updating rules of the form LifeStep3D[{p_, q_, r_}, a_List] := MapThread[If[ #1  1 && p ≤ #2 ≤ q || #2  r, 1, 0]&, {a, Sum[RotateLeft[ a, {i, j, k}], {i, -1, 1}, {j, -1, 1}, {k, -1, 1}] - a}, 3] Carter Bays discovered between 1986 and 1990 the three examples {5, 7, 6} , {4, 5, 5} , and {5, 6, 5} .

The magnetic energy of the system is taken to be e[s_] := -1/2 Apply[Plus, s ListConvolve[ {{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}, s, 2], {0, 1}] so that each pair of adjacent spins contributes -1 when they are parallel and +1 when they are not. … One marginally more complicated case effectively involving 13 neighbors is IsingEvolve[list_, t_Integer] := First[Nest[IsingStep, {list, Mask[list]}, t]] IsingStep[{a_, mask_}] := {MapThread[ If[#2  2 && #3  1, 1 - #1, #1]&, {a, ListConvolve[ {{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}, a, 2], mask}, 2], 1 - mask} where Mask[list_] := Array[Mod[#1 + #2, 2]&, Dimensions[list]] is set up so that alternating checkerboards of cells are updated on successive steps.

Implementation [of 2D cellular automata] An n × n array of white squares with a single black square in the middle can be generated by PadLeft[{{1}}, {n, n}, 0, Floor[{n, n}/2]] For the 5-neighbor rules introduced on page 170 each step can be implemented by CAStep[rule_, a_] := Map[rule 〚 10 - # 〛 &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}] where rule is obtained from the code number by IntegerDigits[code, 2, 10] . For the 9-neighbor rules introduced on page 177 CAStep[rule_, a_] := Map[rule 〚 18 - # 〛 &, ListConvolve[{{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}, a, 2], {2}] where rule is given by IntegerDigits[code, 2, 18] .

One starts by converting the list of cell colors at each step to a polynomial FromDigits[list, x] .

Lengths of [number] representations (a) n , (b) Floor[Log[2, n] + 1] , (c) Tr[FixedPointList[Max[0, Ceiling[Log[2, #]]] &, n + 2]] - n - 3 , (d) 2 Ceiling[Log[3, 2n + 1]] , (e) Floor[Log[GoldenRatio, √ 5 (n + 1/2)]] .

Pages 818 and 1175 discuss the sequence of all Nand theorems listed in order of increasing complexity.

Given an initial condition {i, list, n} the initial condition for the 2-color Turing machine is With[{b = Ceiling[Log[2, k]]}, {i, Flatten[IntegerDigits[list, 2, b]], b n}]