And so, for example, with the setup shown in the main text, rule 54 can emulate rule 0 only with blocks of length b = 6 .

To emulate cellular automaton evolution one starts by encoding a list of cell values by the single combinator p[num[Length[list]]][ Fold[p[Nest[s, k, #2]][#1] &, p[k][k], list]] //. crules where num[n_] := Nest[inc, s[k], n] inc = s[s[k[s]][k]] One can recover the original list by using Extract[expr, Map[Reverse[IntegerDigits[#, 2]] &, 3 + 59(16^Range[Depth[expr[s[k]][s][k] //. crules] - 1, 1, -1] - 1)/ 15)]]/.

In the late 1950s maximal length shift register sequences (page 1084 ) and some error-correcting codes (page 1101 ) were found by systematic searches of possible polynomials.

 [i, j, m], {i, 0, t - 1}, {j, Max[1, n - i], n + i}, {k, 0, ktot - 1}, {m, k + 1, ktot - 1}],  [0, s], Cases[MapIndexed[  [Abs[n - First[#2]], First[#2], #1]&, a],  [x_, _, _] /; x < t], Table[  [Abs[n - i], i, 0], {i, Length[a] + 1, n + t - 1}], Table[!

(OddQ[Length[#]] &), {2}] (see page 358 ) • Flatten[Table[Map[{t, #} &, Fold[Flatten[{#1, #1 + #2}] &, 0, Flatten[2^(Position[ Reverse[IntegerDigits[t, 2]], 1] - 1)]]], {t, 2 n - 1}], 1] (see page 870 ) • Map[Map[FromDigits[#, 2] &, Transpose[Partition[#, 2]]] &, Position[Nest[{{#, #}, {#}} &, 1, n], 1] - 1] (see page 509 ) A formatting hack giving the same visual pattern is DisplayForm[Nest[SubsuperscriptBox[#, #, #] &, "1", n]]

In different systems, the characteristic length used typically in the definition of Reynolds number is different.

Already in 1882 George FitzGerald and Hendrik Lorentz noted that if there was a contraction in length by a factor Sqrt[1 - v 2 /c 2 ] in any object moving at speed v (with c being the speed of light) then this would explain the result.

The total number of possible lists of integers of given finite length—and thus the number of possible rational numbers—turns out also to be ℵ 0 .

Note that if the rule for the finite automaton is represented for example as {{1, 2}, {2, 1}} where each sublist corresponds to a particular state, and the elements of the sublist give the successor states with inputs Range[0, k - 1] , then the n th element in the output sequence can be obtained from Fold[rule 〚 #1, #2 〛 &, 1, IntegerDigits[n - 1, k] + 1] - 1 while the first k m elements can be obtained from Nest[Flatten[rule 〚 # 〛 ] &, 1, m] - 1 To treat examples such as case (c) where elements can subdivide into blocks of several different lengths one must generalize the notion of digit sequences.

The smallest solution for x is given by Numerator[FromContinuedFraction[ ContinuedFraction[ √ a , (If[EvenQ[#], #, 2 #] &)[ Length[Last[ContinuedFraction[ √ a ]]]]]]] This is plotted below; complicated variation and some very large values are seen (with a = 61 for example x  1766319049 ).