The Perrin sequence f[n_] := f[n - 2] + f[n - 3] ; f[0] = 3; f[1] = 0 ; f[2] = 2 has the peculiar property that Mod[f[n], n]  0 mostly but not always only for n prime.

The code numbers in these cases are given by 2/3 (4 n - 1) + Apply[Plus, 4 list ] where n is the number of neighbors, here 5.

And from Euler's formula f + n = e + 2 , it then follows that the average number of edges of each face is always 6(1 - 2/f) , where f is the total number of faces.

In class 1, changes always die out, and in fact exactly the same final state is reached regardless of what initial conditions were used.

But what about the number whose n th digit is 1 - f[n, n] ? … But what about the program with output 1 - f[n, n] ?

In the mid-1800s it became clear that despite their different origins most of these functions could be viewed as special cases of Hypergeometric2F1[a, b, c, z] , and that the functions covered the solutions to all linear differential equations of a certain type. ( Zeta and PolyLog are parametric derivatives of Hypergeometric2F1 ; elliptic modular functions are inverses.) … to be Gamma[x + 1] .)

(Already in the late 1940s John von Neumann had suggested using x  4x (1 - x) as a random number generator, commenting on its extraction of initial condition digits, as mentioned on page 921 .) Some detailed analytical studies of logistic maps of the form x  a x (1 - x) were done in the late 1950s and early 1960s—and in the mid-1970s iterated maps became popular, with much analysis and computer experimentation on them being done. … In connection with his study of continued fractions Carl Friedrich Gauss noted in 1799 complexity in the behavior of the iterated map x  FractionalPart[1/x] .

If one looks at the history of a single row of cells, it typically looks much like the complete histories we have seen in 1D class 4 cellular automata. … The so-called "switch engine" discovered in 1971 generates unbounded growth by leaving a trail behind when it moves; it is now known that it can be obtained from an initial condition with 10 black cells, or black cells in just a 5×5 or 39×1 region. … In that case, the evolution is effectively 1D, and turns out to follow elementary rule 22, thus producing the infinitely growing nested pattern shown on page 263 .

It can then be decoded as PowerMod[c, e, n] , where e = PowerMod[d, -1, EulerPhi[n]] .

Probabilistic models A probabilistic model must associate with every sequence a probability that is a number between 0 and 1.