Doubling rules [cellular automata] Rule (a) is {{0, 2, _}  5, {5, 3, _}  5, {5, _, _}  1, {_, 5, _}  1, {_, 2, _}  3, {_, 3, 2}  2, {_, 1, 2}  4, {_, 4, _}  3, {4, 3, _}  4, {4, 0, _}  2, {_, x_, _}  x} and takes 2n 2 + n steps to yield Table[1, {2n}] given input Append[Table[1, {n - 1}], 2] .

Cyclic tag systems which allow any value for each element can be obtained by adding the rule CTStep[{{r_, s___}, {n_, a___}}] := {{s, r}, Flatten[{a, Table[r, {n}]}]} The leading elements in this case can be obtained using CTListStep[{rules_, list_}] := {RotateLeft[rules, Length[list]], With[{n = Length[rules]}, Flatten[Apply[Table[#1, {#2}] &, Map[Transpose[ {rules, #}] &, Partition[list, n, n, 1, 0]], {2}]]]}

As one example, consider a large number of circular coins pushed together on a table. … And it turns out that in this particular case this pattern is quickly produced if one actually pushes coins together on a table.

Note that in practice a coin tossed in the air will typically turn over between ten and twenty times while a die rolled on a table will turn over a few tens of times. A coin spun on a table can rotate several hundred times before falling over and coming to rest.

[Network] cluster numbers The following tables give the total numbers of distinct clusters—with number of nodes going across the page, and number of dangling connections going down.

Cyclic tag systems [emulating tag systems] From a tag system which depends only on its first element, with rules given as in the note below, the following constructs a cyclic tag system emulating it: TS1ToCT[{n_, subs_}] := With[{k = Length[subs]}, Join[Map[v[Last[#], k] &, subs], Table[{}, {k(n - 1)}]]] u[i_, k_] := Table[If[j  i + 1, 1, 0], {j, k}] v[list_, k_] := Flatten[Map[u[#, k] &, list]] The initial condition for the tag system can be converted using v[list, k] .

[Sequences with] flat spectra Any impulse sequence Join[{1}, Table[0, {n}]] will yield a flat spectrum. … Sequences of 0's and 1's that have the same property are {1, 0, 1, 0} , {1, 0, 0, 1, 0, 0, 1, 0, 0} or in general Flatten[Table[{1, Table[0, {n - 1}]}, {n}]] .

{i[r_]  Table[n + j p  (1 + n Prime[r] (-n + #) &), {j, 0, g - 1}], d[r_, k_]  Table[n + j p  If[Mod[j, Prime[r]]  0, -1 + k + (-n + #)/Prime[r] &, # + 1 &], {j, 0, g - 1}]}] &, prog]]]}] The rules for the arithmetic system are represented so that the system from page 122 becomes for example {2, {0  (3 #/2 &), 1  (3 (# + 1)/2 &)}} . If the register machine starts at instruction n with values regs in its registers, then the corresponding arithmetic system starts with the number n + Table[Prime[i]^reg 〚 i 〛 , {i, nr}] p - 1 where p = Length[prog] .

But if there are an infinite number of elements that can be specified in the initial condition—as in a cellular automaton—then a table for an oracle could also be given in the initial conditions.

In general, with n elements f can be specified by an n × n "multiplication table". … Pictures (a) and (b) below however correspond to the n = 3 multiplication tables {{1, 1, 3}, {3, 3, 2}, {2, 2, 1}} and {{3, 1, 3}, {1, 3, 1}, {3, 1, 2}} . Note that for (b) the table is symmetric, corresponding to a commutative multiplication operation.