An explicit table of 40,000 random digits was created in 1927 by Leonard Tippett from details of census data. And in 1938 further tables were generated by Ronald Fisher from digits of logarithms. Several tables based on physical processes were produced, with the RAND Corporation in 1955 publishing a table of a million random digits obtained from an electronic roulette wheel.

Numbers of possible [2D cellular automaton] rules The table below gives the total number of 2D rules of various types with two possible colors for each cell. … Rules are considered rotationally symmetric in the table below if they preserve any possible rotational symmetry consistent with the underlying arrangement of cells.

Rule expressions [for cellular automata] The table below gives Boolean expressions for each of the elementary rules.

In this book such offset lists are always taken to be in the order given by Sort , so that for range r rules in d dimensions the order is the same as Flatten[Array[List, Table[2r + 1, {d}], -r], d - 1] . … With offset list os and k colors the possible neighborhood configurations are Reverse[Table[IntegerDigits[i - 1, k, Length[os]], {i, k^Length[os]}]] (These are shown on page 53 for elementary rules and page 941 for 5-neighbor rules.) … A single step in evolution of a general cellular automaton with state a and rule number num is then given by Map[IntegerDigits[num, k, k^Length[os]] 〚 -1 - # 〛 &, Apply[Plus, MapIndexed[k^(Length[os] - First[#2]) RotateLeft[a, #1] &, os]], {-1}] or equivalently by Map[IntegerDigits[num, k, k^Length[os]] 〚 -# - 1 〛 &, ListCorrelate[Fold[ReplacePart[k #1, 1, #2 + r + 1] &, Array[0 &, Table[2r + 1, {d}]], os], a, r + 1], {d}]

Random walks In one dimension, a random walk with t steps of length 1 starting at position 0 can be generated from NestList[(# + (-1)^Random[Integer])&, 0, t] or equivalently FoldList[Plus, 0, Table[(-1)^Random[Integer], {t}]] A generalization to d dimensions is then FoldList[Plus, Table[0, {d}], Table[RotateLeft[PadLeft[ {(-1)^Random[Integer]}, d], Random[Integer, d - 1]], {t}]] A fundamental property of random walks is that after t steps the root mean square displacement from the starting position is proportional to √ t . … To make a random walk on a lattice with k directions in two dimensions, one can set up e = Table[{Cos[2 π s/k], Sin[2 π s/k]}, {s, 0, k - 1}] then use FoldList[Plus, {0, 0}, Table[e 〚 Random[Integer, {1, k}] 〛 , {t}]] It turns out that on any regular lattice, in any number of dimensions, the average behavior of a random walk is always isotropic.

Matrices satisfying constraints One can consider for example magic squares, Latin squares (quasigroup multiplication tables), and matrices having the Hadamard property discussed on page 1073 .

The operator forms shown can be thought of as giving multiplication tables.

But the table below demonstrates that as soon as one goes beyond the familiar traditions of language and mathematics there are other operators that can also just as well be used as primitives.

Pointer-based encoding One can encode a list of data d by generating pointers to the longest and most recent copies of each subsequence of length at least b using PEncode[d_, b_ : 4] := Module[{i, a, u, v}, i = 2; a = {First[d]}; While[i ≤ Length[d], {u, v} = Last[Sort[Table[{MatchLength[d, i, j], j}, {j, i - 1}]]]; If[u ≥ b, AppendTo[a, p[i - v, u]]; i += u, AppendTo[a, d 〚 i 〛 ]; i++]]; a] MatchLength[d_, i_, j_] := With[{m = Length[d] - i}, Catch[ Do[If[d 〚 i + k 〛 =!… One can reproduce the original data using PDecode[a_] := Module[{d = Flatten[ a /. p[j_, r_]  Table[p[j], {r}]]}, Flatten[MapIndexed[ If[Head[#1] === p, d 〚 #2 〛 = d 〚 #2 - First[#1] 〛 ,#1] &, d]]] To get a representation purely in terms of 0 and 1, one can use a self-delimiting representation for each integer that appears. … An example is the sequence Table[Append[Table[0, {r}], 1], {r, s}] whose encoded version grows like √ n Log[n] .

.}] = 0 g[{1, s__}] := 1 + g[IntegerDigits[FromDigits[{s}, 2] + 1, 2]] The list of elements in the sequence up to value m is given by Flatten[Table[Table[n, {IntegerExponent[n, 2] + 1}], {n, m}]] The differences between the first 2 (2 k -1) of these elements is Nest[Replace[#, {x___}  {x, 1, x, 0}]&, {}, k] The largest n for which f[n]  m is given by 2m + 1 - DigitCount[m, 2, 1] or IntegerExponent[(2m)!… 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.