It exhibits a nested structure, and can be obtained as in the pictures below from the evolution of a 2D substitution system, or equivalently from a Kronecker product as in Nest[Flatten2D[Map[# {{1, 1}, {1, -1}} &, #, {2}]] &, {{1}}, s] with Flatten2D[a_] := Apply[Join, Apply[Join, Map[Transpose,a], {2}]] (c) is known as dyadic or Paley order. It is related to (a) by Gray code reordering of the rows, and to (b) by reordering according to (see page 905 ) BitReverseOrder[a_] := With[{n = Length[a]}, a 〚 Map[FromDigits[Reverse[#], 2] &, IntegerDigits[Range[0, n - 1], 2, Log[2, n]]] + 1 〛 ] It is also given by Array[Apply[Times, (-1)^(IntegerDigits[#1, 2, s] Reverse[IntegerDigits[#2, 2, s]])] &, 2^{s,s}, 0] where (b) is obtained simply by dropping the Reverse . … The function at position 2/3 (1 + 4^-(Floor[s/2] + 1/2)) 2 s in basis (a), for example, is exactly the Thue–Morse sequence (with 0 replaced by -1) from page 83 .

And what this means is that instead of depending only on quantities like f[n – 1] and f[n – 2] , the rule for f[n] can also for example depend on a quantity like f[n – f[n – 1]] . … And if it is not, then results obtained by applying the rule can involve meaningless quantities such as f[0] , f[–1] and f[–2] . Examples of sequences generated by rules that do not depend only on elements a fixed distance back.

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. … Note that any sequence of 0's and 1's corresponds to the beginning of the encoding for some sequence or another. … The overall fraction of a length n input that consists of repeats of length at least b is greater than 1 - 2 b /n and is essentially 1 - Sum[(1 - 2 -b ) i Product[1 + (1 - 2 -b ) j - (1 - 2 -b - 1 ) j , {j, i - b + 1, i - 1}], {i, b, n - b}]/(n - 2b + 1)

Reynolds numbers If a system is to act like a continuum fluid, then almost by definition its behavior can involve only a limited number of macroscopic quantities, such as density and velocity. … For water, ν = 0.01 , for air ν = 0.15 , and for glycerine ν = 10000 , all in units of cm 2 /sec . … But the fact that the form of flow should depend only on Reynolds number means that in the pictures in the main text for example it is not necessary to specify absolute sizes or speeds: one need only know the product U L that appears in the Reynolds number.

The first 2 m elements in the sequence can be obtained from (see page 1081 ) (CoefficientList[Product[1 - z 2 s , {s, 0, m - 1}], z] + 1)/2 The first n elements can also be obtained from (see page 1092 ) Mod[CoefficientList[Series[(1 + Sqrt[(1 - 3x)/(1 + x)])/ (2(1 + x)), {x, 0, n - 1}], x], 2] The sequence occurs many times in this book; it can for example be derived from a column of values in the rule 150 cellular automaton pattern discussed on page 885 . (c) (Fibonacci-related sequence) The sequence at step t can be obtained from a[t_] := Join[a[t - 1], a[t - 2]]; a[1] = {0}; a[2] = {0, 1} . … (d) (Cantor set) The color of the element at position n is given by If[FreeQ[IntegerDigits[n - 1, 3], 1], 1, 0] , which turns out to be equivalent to If[OddQ[n], Sign[Mod[Binomial[n - 1, (n - 1)/2], 3]], 0, 1] There are 3 t elements after t steps, of which 2 t are black.

Given three cells {a 1 , a 2 , a 3 } the rule specifies that the new value of the center cell will be Mod[a 1 + a 3 , 2] . But given {a 1 , 0, a 2 , 0, a 3 , 0} the value after one step is {Mod[a 1 + a 2 , 2], 0, Mod[a 2 + a 3 , 2], 0} and after two steps is again {Mod[a 1 + a 3 , 2], 0} . … If k is not prime the pattern is no longer strictly invariant with respect to keeping only every k th row and column—but is in effect still a superposition of patterns with this property for factors of k .

Some integer functions can readily be obtained by supplying integer arguments to continuous functions, so that for example Mod[x, 2] corresponds to Sin[ π x/2] 2 or (1 - Cos[ π x])/2, Mod[x, 3] ↔ 1 + 2/3(Cos[2/3 π (x - 2)] - Cos[2 π x/3]) Mod[x, 4] ↔ (3 - 2 Cos[ π x/2] - Cos[ π x] - 2 Sin[ π x/2])/2 Mod[x, n] ↔ Sum[j Product[(Sin[ π (x - i - j)/n]/ Sin[ π i/n]) 2 , {i, n - 1}], {j, n - 1}] (As another example, If[x > 0, 1, 0] corresponds to 1 - 1/Gamma[1 - x] .) And in this way the discrete system x  If[EvenQ[x], 3x/2, 3(x + 1)/2] from page 122 can be emulated by the continuous iterated map x  (3 + 6 x - 3 Cos[ π x])/4 . … Page 147 showed how Sin[x] + Sin[ √ 2 x] has nested features, and these are reflected in the distribution of eigenvalues for ODEs containing such functions.

The Turing machine {{1, 2}  {2, 2, -1}, {1, 1}  {1, 1, -1}, {1, 0}  {3, 1, 1}, {2, 2}  {4, 0, -1}, {2, 1}  {1, 2, -1}, {2, 0}  {2, 1, -1}, {3, 2}  {3, 2, 1}, {3, 1}  {3, 1, 1}, {3, 0}  {1, 0, -1}, {4, 2}  {2, 2, 1}, {4, 1}  {4, 1, 1}, {4, 0}  {2, 2, -1}} with s = 4 states and k = 3 possible colors also emulates rule 110 when started from Prepend[list + 1, 1] surrounded by 0 's. The s = 3 , k = 4 Turing machine {{1 , 0}  {1, 2, 1}, {1, 1}  {2, 3, 1}, {1, 2}  {1, 0, -1}, {1, 3}  {1, 1, -1}, {2, 0}  {1, 3, 1}, {2, 1}  {3, 3, 1}, {3, 0}  {1, 3, 1}, {3, 1}  {3, 2, 1}} started from Append[list, 0] with 0 's on the left and 2 's on the right generates a shifted version of rule 110. Note that this Turing machine requires only 8 out of the 12 possible cases in its rules to be specified.

With odd n the same turns out to be true for sequences Exp[2 π  Mod[Range[n] 2 , n]/n] —a fact used in the design of acoustic diffusers (see page 1183 ). For sequences involving only two distinct integers flat spectra are rare; with ± 1 those equivalent to {1, 1, 1, -1} seem to be the only examples. ( {r 2 , r s, s 2 , -r s} works for any r and s , as do all lists obtained working modulo x n - 1 from p[x]/p[1/x] where p[x] is any invertible polynomial.) … 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}]] .

In studying specific instances of objects like groups one often represents elements as products of constants or generators, and then for example specifies the group by giving relations between these products.