All rules of the kind shown here lead to sequences where f[n] can be expressed in terms of a simple sum of powers of the form a n .

Whenever a number x is known to satisfy Sum[a[i] f[i][x], {i, n}]  0 with fixed f[i] one can take the early digits of x and use LatticeReduce to find integer solutions for the a[i] .

Each step in its evolution can be implemented using LifeStep[a_List] := MapThread[If[(#1  1 && #2  4) || #2  3, 1, 0]&, {a, Sum[RotateLeft[a, {i, j}], {i, -1, 1}, {j, -1, 1}]}, 2] A more efficient implementation can be obtained by operating not on a complete array of black and white cells but rather just on a list of positions of black cells.

The number of strings of depth d (and thus taking d steps to annihilate completely) is given by c[{n, n}, d] - c[{n, n}, d - 1] where c[{_, _}, -1] = 0; c[{0, 0}, _] = 1; c[{m_, n_}, _] := 0 /; n > m; c[{m_, n_}, d_] := Sum[c[{i, j}, d], {i, 0, m - 1}, {j, m - d, n - 1}] Several types of structures are equivalent to strings of balanced parentheses, as illustrated below.

Gaussian distributions typically arise when measurements involve sums of random quantities; other common distributions are obtained from products or other simple combinations of random quantities, or from the results of simple processes based on random quantities.

In base k a number is constructed from a digit sequence a[r] , … , a[1] , a[0] (with 0 ≤ a[i] < k ) according to Sum[a[i] k i , {i, 0, r}] . But given a sequence of digits that are each 0 or 1, it is also possible for example to construct numbers according to Sum[a[i] Fibonacci[i + 2], {i, 0, r}] .

And in the late 1980s, building on work of mine from 1984 (described on page 276 ), James Crutchfield made a study of such models in which he defined the complexity of a model to be equal to -p Log[p] summed over all connections in the network.

And third, that the so-called triangle inequality holds, so that the distance AC is no greater than the sum of the distances AB and BC .

Thus for example, if there are two different ways that some process can occur, it suggests that the total probability for the whole process should be just the sum of the probabilities for the process to occur in the two different ways.

The formula consists of a sum of terms, with each term being zero unless the colors of the three cells match a situation in which the rule yields a black cell.