Thus for example {(0 | 1) ...} corresponds to all possible sequences of 0 's and 1 's, while {1, 1, (1) ..., 0, (0) ...} ... corresponds to the sequences that can occur after 2 steps in rule 126 and {(0) ..., 1, {0, (0) ..., 1, 1} | {1, (1) ..., 0}} ... to those that can occur after 2 steps in rule 110 (see page 279 ).

But while iteration is generally viewed as being quite easy to understand, until recently even recursion was usually considered rather difficult.

Note that in case (a), the total number of possible states at step t increases roughly like t 2 , while in case (b) it increases only like t .

And while some such questions may be answered by fairly straightforward computational or mathematical means, there will be no upper bound on the amount of effort that it could take to answer any particular question.

Results about primes Prime[n] is given approximately by n Log[n] + n Log[Log[n]] . ( Prime[10 9 ] is 22,801,763,489 while the approximation gives 2.38 × 10 10. ) A first approximation to PrimePi[n] is n/Log[n] . … This was found empirically by Carl Friedrich Gauss in 1792, based on looking at a table of primes. ( PrimePi[10 9 ] is 50,847,534 while LogIntegral[10 9 ] is about 50,849,235.)

For while it is easy to tell that a cave painting of an animal is a piece of purposeful art, dots carved into a rock in an approximate rule 30 pattern might not even be noticed as something of human origin.

BesselJ[0, x] goes like Sin[x]/ √ x for large x while AiryAi[-x] goes like Sin[x 3/2 ]/x 1/4 .

The top pictures show step 30,000, while those on the bottom right show step 200.

But on the first page , the initial conditions are set up so as to make the universal cellular automaton emulate rule 254, while on the second page they are set up to make it emulate rule 90, and on the third page rule 30.

With a concentrations list c , the position p of a new element is given by Position[c, Max[c], 1, 1] 〚 1, 1 〛 , while the new list of concentrations is λ c + RotateRight[f, p] where f is a list of depletions associated with addition of a new element at position 1.