Fairly large terms are sometimes seen quite early: in 5 1/3 term 19 is 3052, while in Root[10 + 8 # - # 3 &, 1] term 34 is 1,501,790. … For any irrational number this quantity cannot be less than 2, while for algebraic irrationals Klaus Roth showed in 1955 that it can only have finitely many peaks that reach above any specified level.

Pointer-based encoding One can encode a list of data d by generating pointers to the longest and most recent copies of each subsequence of length at least b using PEncode[d_, b_ : 4] := Module[{i, a, u, v}, i = 2; a = {First[d]}; While[i ≤ Length[d], {u, v} = Last[Sort[Table[{MatchLength[d, i, j], j}, {j, i - 1}]]]; If[u ≥ b, AppendTo[a, p[i - v, u]]; i += u, AppendTo[a, d 〚 i 〛 ]; i++]]; a] MatchLength[d_, i_, j_] := With[{m = Length[d] - i}, Catch[ Do[If[d 〚 i + k 〛 =!

But while this axiom is convenient in simplifying work in set theory it has not been found generally useful in mathematics, and is normally considered optional at best.

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.

(To recover correct infinite size results one must increase size while keeping the number of steps of evolution fixed; the networks shown above, however, effectively depend on arbitrarily many steps of evolution.)

Often this happens quickly, but sometimes all three bodies show complex and apparently random behavior for quite a while.

And while its more elementary aspects are certainly crucial for everyday modern life, beyond basic algebra its central place in education must presumably be justified more on the basis of promoting overall patterns of thinking than in supplying specific factual knowledge of everyday relevance.

In case (b), the number of steps is equal to the number of base 2 digits in n , while in case (c) it is determined by the number of 1's in the base 2 digit sequence of n .

General recursive functions, however, also allow μ [f_] = NestWhile[# + 1 &, 0, Function[n, f[n, ##] ≠ 0]]& which can perform unbounded searches. … It is inevitable that the function shown must eventually grow faster than any primitive recursive function (at x = 356 its value is 63190, while at x = 1464 it is 1073844).

As suggested by the pictures in the main text, spectra such as (b) and (d) in the limit consist purely of discrete Dirac delta function peaks, while spectra such as (a) and (c) also contain essentially continuous parts.