And if this were the case, then the digits of an initial condition could for example be the table for an oracle of the kind discussed on page 1126 —and even a simple shift mapping could then yield output that is computationally more sophisticated than any standard discrete system.

If the initial numbers in the two registers are a and b , then the initial conditions for the cellular automaton are Join[Table[m + 2, {a}], {1}, Table[m + 5,{b}]] surrounded by 0's.

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.

Thus the patterns on page 189 can be formed from t -digit integers in base  - 1 containing only digits 0 and 1, as given by Table[FromDigits[IntegerDigits[s, 2, t],  - 1], {s, 0, 2 t -1}] In the particular case of base  - q with digits 0 through q 2 , it turns out that for sufficiently large t any complex integer can be represented, and will therefore be part of the pattern.

[Rules for the] squaring cellular automaton The rules are {{0, _, 3}  0, {_, 2, 3}  3, {1, 1, 3}  4, {_, 1, 4}  4, {1 | 2, 3, _}  5,{p : (0 | 1), 4, _}  7 - p, {7, 2, 6}  3, {7, _, _}  7, {_, 7, p : (1 | 2)}  p, {_, p : (5 | 6), _}  7 - p, {5 | 6, p : (1 | 2), _}  7 - p, {5 | 6, 0, 0}  1, {_, p : (1 | 2), _}  p, {_, _, _}  0} and the initial conditions consist of Append[Table[1, {n}], 3] surrounded by 0 's.

Runs of digits [in numbers] One can consider any base 2 digit sequence as consisting of successive runs of 0's and 1's, constructed from the list of run lengths by Fold[Join[#1, Table[1 - Last[#1], {#2}]] &, {0}, list] This representation is related to so-called surreal numbers (though with the first few digits different).

The table on the next page gives the continued fraction representations for various numbers.

One can characterize the symmetry of a pattern by taking the list v of positions of cells it contains, and looking at tensors of successive ranks n : Apply[Plus, Map[Apply[Outer[Times, ##] &, Table[#, {n}]] &, v]] For circular or spherical patterns that are perfectly isotropic in d dimensions these tensors must all be proportional to (d - 2)!!… &, Table[Count[{##}, i], {i, d}]]]&, Table[d, {n}]]/(d + n - 2)!!

A logarithmic law for leading digits is also found in many practical numerical tables, as noted by Simon Newcomb in 1881 and Frank Benford in 1938.