Particularly dramatic are the concatenation systems discussed on page 913 , as well as successive rows in nested patterns such as Flatten[IntegerDigits[NestList[BitXor[#, 2 #] &, 1, 500], 2]] and sequences based on numbers such as Flatten[Table[If[GCD[i, j]  0, 1, 0], {i, 1000}, {j, i}]] (see page 613 ).

If Mod[p, 4]  1 JacobiSymbol sequences also satisfy Fourier[list]  list .

The evolution of the arithmetic system is given by ASEvolveList[{n_, rules_}, init_, t_] := NestList[(Mod[#, n] /. rules)[#] &, init, t] Given a value m obtained in the evolution of the arithmetic system, the state of the register machine to which it corresponds is {Mod[m, p] + 1, Map[Last, FactorInteger[ Product[Prime[i], {i, nr}] Quotient[m, p]]] - 1} Note that it is possible to have each successive step involve only multiplication, with no addition, at the cost of using considerably larger numbers overall.

To see the consequences of such constraints consider breaking a sequence of colors into blocks of length n , with each block overlapping by n - 1 cells with its predecessor, as in Partition[list, n, 1] . If all possible sequences of colors were allowed, then there would be k possibilities for what block could follow a given block, given by Map[Rest, Table[Append[list, i], {i, 0, k - 1}]] .

Implementation [of basic aggregation model] One way to represent a cluster is by giving a list of the coordinates at which each black cell occurs. … With a grid of cells set up in advance, each step in this type of Eden model can be achieved with AStep[a_] := ReplacePart[a, 1, (# 〚 Random[ Integer, {1, Length[#]}] 〛 &)[Position[(1 - a)Sign[ ListConvolve[{{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}, a, {2, 2}]], 1]]] This implementation can readily be extended to generalized aggregation models (see below ).

The key idea is to represent data of any kind by a symbolic expression of the general form head[arg 1 , arg 2 , …] . ( a + b 2 is thus Plus[a, Power[b, 2]] , {a, b, c} is List[a, b, c] and a = b + 1 is Set[a, Plus[b, 1]] .)

Often these can be thought of as one-way versions of axioms for operator systems (see page 1172 ), but applied only once per step (as /. does), rather than in all possible ways (as in a multiway system)—so that the evolution is just given by NestList[#/.rule &, init, t] .

Starting with an ordinary base 2 digit sequence, one prepends a unary specification of its length, then a specification of that length specification, and so on: (Flatten[{Sign[-Range[1 - Length[#], 0]], #}] &)[ Map[Rest, IntegerDigits[Rest[Reverse[NestWhileList[ Floor[Log[2, #] &, n + 1, # > 1 &]]],2]]] (d) Binary-coded base 3. … Apply[Take, RealDigits[(N[#, N[Log[10, #] + 3]] &)[ n √ 5 /GoldenRatio 2 + 1/2], GoldenRatio]] The representations of all the first Fibonacci[n] - 1 numbers can be obtained from (the version in the main text has Rest[RotateLeft[Join[#, {0, 1}]]] & applied) Apply[Join, Map[Last, NestList[{# 〚 2 〛 ], Join[Map[Join[{1, 0}, Rest[#]] & , # 〚 2 〛 ], Map[Join[{1, 0}, #] &, # 〚 1 〛 ]]} &, {{}, {{1}}}, n-3]]]

But even among hydrocarbons with fairly few atoms not all have ever been considered interesting enough to list in standard chemical databases. Thus for example the following compares the total number of conceivable alkanes (paraffins) to the number actually listed in the 2001 standard Beilstein database: Any tree with up to 4 connections at each node can in principle correspond to an alkane with chemical formula C n H 2n + 2 .

And with this setup, t steps of evolution can be found with SSSEvolveList[rule_, init_s, t_Integer] := NestList[(# /. rule)&, init, t] Note that as an alternative to having s be Flat , one can explicitly set up rules based on patterns such as s[x___, 1, 0, y___]  s[x, 0, 1, 0, y] .