The digits that lie directly below and to the left of the original 1 at the top of the pattern correspond to the whole number part of each successive number (e.g. 3 in 3.375), while the digits that lie to the right correspond to the fractional part (e.g. 0.375 in 3.375).

So long as f[n] grows less rapidly than 2 n (as when f = Fibonacci or f = Prime ), digits 0 and 1 will suffice, though the representation is not generally unique.

A sequence of much faster methods have however been developed over the past few decades, one simple example that works for most n being the so-called rho method of John Pollard (compare the quadratic residue sequences discussed below): Module[{f = Mod[# 2 + 1, n] &, a = 2, b = 5, c}, While[(c = GCD[n, a - b])  1, {a, b} = {f[a], f[f[b]]}]; c] Most existing methods depend on facts in number theory that are fairly easy to state, though implementing them for maximum efficiency tends to lead to complex programs.

The pictures below show how 1D cellular automata can be implemented in the 4-color WireWorld cellular automaton of Brian Silverman from 1987, whose rules find the new value of a cell from its old value a and the number u of its 8 neighbors that are 1's according to a /. {0  0, 1  2, 2  3, 3  If[0 < u < 3, 1, 3]}

Digit reversal Sequences of the form Table[FromDigits[ Reverse[IntegerDigits[n, k, m]], k], {n, 0, k m - 1}] shown below appear in algorithms such as the fast Fourier transform and, with different values of k for different coordinates, in certain quasi-Monte Carlo schemes.

Formal languages [and constraints] Formal languages of the kind discussed on page 938 can be used to define constraints on 1D sequences.

The issue of which sequence of horizontal and vertical sides the ball hits depends on the exact slope with which the ball is started (in the picture below it is 1/ √ 2 ).

One feature often found is that the average radius of "droplets" increases with time roughly like t 1/3 .

And with the constraint of reversibility, it turns out that it is impossible to get a non-trivial phase transition in any 1D system with the kind of short-range interactions that exist in a cellular automaton. But in systems whose evolution is not reversible, it is possible for phase transitions to occur in 1D, as the examples in the main text show. … When the total number of cells increases, however, the fraction of such configurations rapidly decreases, and in the infinite size limit, there are no such configurations, and a truly discontinuous transition occurs exactly at density 1/2.

Note that GCD[m, n] yields a more complicated pattern (see page 613 ), as do JacobiSymbol[m, 2n - 1] (see page 1081 ) and various combinations of functions (see page 747 ).