At stage n the number of polyominoes of each type is Fibonacci[2n - {2, 0, 1}]/{1, 2, 1} .

Periods from 1 to 15 are represented by different rows, with period 1 at the bottom. … The total number of configurations in rule 30 that repeat with periods that divide 1 through 10 are {3, 3, 15, 10, 8, 99, 18, 14, 30, 163} . … For p = 2 , rule 18 leaves 20 of the 32 possible length 5 blocks invariant, but these blocks can only be strung together to yield repetitions of {a, b, 0, 0} , where now a and b are not fixed, but in every case can each be either {1} or {0, 1} .

In the first case shown, this number varies like (1/a) 1 for small a , while in the last case, it varies like (1/a) 2 . In general, if the number varies like (1/a) d , one can take d to be the dimension of the pattern. … But even when this does not happen, the limiting behavior for small a is still (1/a) d for any nested pattern.

One can take the original stem to extend from the point -1 to 0; the rule is then specified by the list b of complex numbers corresponding to the positions of the new tip obtained after one step. And after n steps the positions of all tips generated are given simply by Nest[Flatten[Outer[Times, 1 + #, b]] &, {0}, n]

Generating causal networks If every element generated in the evolution of a generalized substitution system is assigned a unique number, then events can be represented for example by {4, 5}  {11, 12, 13} —and from a list of such events a causal network can be built up using With[{u = Map[First, list]}, MapIndexed[Function[ {e, i}, First[i]  Map[(If[# === {}, ∞ , # 〚 1, 1 〛 ] &)[ Position[u, #]]) &, Last[e]]], list]]

Zipf's law To a fairly good approximation the n th most common word in a large sample of English text occurs with frequency 1/n , as illustrated in the first picture below. … When k = 1 , the n th most common word will have frequency c -n . … The normalization of probabilities then implies p = 1/(2k) , and since the word at rank roughly k m then has probability 1/(2k) m , Zipf's law follows.

But the number of diagrams grows rapidly with order, and in fact the k th order term can be about (-1) k α k (k/2)! … (The high-order terms often seem to be associated with asymptotic series for things like Exp[-1/ α ] .) … Ignoring parts that depend on particle masses the result (derived in successive orders from 1, 1, 7, 72, 891 diagrams) is 2 ( 1 + α /(2 π ) + (3 Zeta[3]/4 - 1/2 π 2 Log[2] + π 2 /12 + 197/144) ( α / π ) 2 + (83/72 π 2 Zeta[3] - 215 Zeta[5]/24 - 239 π 4 /2160 + 139 Zeta[3]/18 + 25/18 (24 PolyLog[ 4, 1/2] + Log[2] 4 - π 2 Log[2] 2 ) - 298/9 π 2 Log[2] + 17101 π 2 /810 + 28259/5184) ( α / π ) 3 - 1.4 ( α / π ) 4 + …), or roughly 2. + 0.32 α - 0.067 α 2 + 0.076 α 3 - 0.029 α 4 + … The comparative simplicity of the symbolic forms here (which might get still simpler in terms of suitable generalized polylogarithm functions) may be a hint that methods much more efficient than explicit Feynman diagram evaluation could be used.

In most cases the number of steps to generate the final pattern increases roughly linearly with the width of the input—although in the case of the fourth-to-last rule on the second row it is 2(n 2 -n+1) for width n .

Digit count sequences Starting say with {1} repeatedly replace list by Join[list, IntegerDigits[Apply[Plus, list], 2]] The resulting sequences grow in length roughly like n Log[n] . The picture below shows the fluctuations around m/2 of the cumulative number of 1's up to position m in the sequence obtained at step 1000.

Conway considered fraction systems based on rules of the form FSEvolveList[fracs_, init_, t_] := NestList[First[Select[fracs #, IntegerQ, 1]] &, init, t] With the choice fracs = {17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/ 23, 77/19, 1/17, 11/13, 13/11, 15/14, 15/2, 55/1} starting at 2 the result for Log[2, list] is as shown below, where Rest[Log[2, Select[list, IntegerQ[Log[2, #]] &]]] gives exactly the primes.