Implementation [of patterning model] Given a 2D array of values a and a list of weights w , each step in the evolution of the system corresponds to WeightedStep[w_List, a_] := Map[If[# > 0, 1, 0]&, Sum[w 〚 1 + i 〛 Apply[Plus, Map[RotateLeft[a, #]&, Layer[i]]], {i, 0, Length[w] - 1}], {2}] Layer[n_] := Layer[n] = Select[Flatten[Table[{i, j}, {i, -n, n}, {j, -n, n}],1], MemberQ[#, n| - n]&]

With two possible forms of behavior h[i] = 0 or 1 for initial condition i , an example of such a number is Sum[2 -i h[i], {i, ∞ }] . Closely related is the total probability for each form of behavior, given for example by Sum[2^-(Ceiling[Log[2, i]]) h[i], {i, ∞ }] .

Instead of decomposing a number into a sum of powers of an integer base, one decomposes it into a sum of Fibonacci numbers (see page 902 ).

The so-called measure entropy is given as discussed on page 959 by the limit of -Sum[p i Log[k, p i ], {i, k b }]/b where the p i are the probabilities for the blocks.

Iterated aliquot sums Related to case (b) above is a system which repeats the replacement n  Apply[Plus, Divisors[n]] - n or equivalently n  DivisorSigma[1, n] - n .

Substitution systems [and sine sums] Cos[a x] - Cos[b x] has two families of zeros: 2 π n/(a + b) and 2 π n/(b - a) .

3D class 4 [cellular automaton] rules With a cubic lattice of the type shown on page 183 , and with updating rules of the form LifeStep3D[{p_, q_, r_}, a_List] := MapThread[If[ #1  1 && p ≤ #2 ≤ q || #2  r, 1, 0]&, {a, Sum[RotateLeft[ a, {i, j, k}], {i, -1, 1}, {j, -1, 1}, {k, -1, 1}] - a}, 3] Carter Bays discovered between 1986 and 1990 the three examples {5, 7, 6} , {4, 5, 5} , and {5, 6, 5} .

The zeta function Zeta[s] is defined as Sum[1/k s , {k, ∞ }] .

The formulas we have used so far can be thought of as always consisting of sums of products of variables.

At a mathematical level, following work by Joseph Fourier around 1810 it became clear by the mid-1800s how any sufficiently smooth function could be decomposed into sums of sine waves with frequencies corresponding to successive integers.