Counting of [network] nodes The number of nodes reached by going out to network distance r (with r > 1 ) from any node in the networks on page 477 is (a) 4r - 4 , (b) 3r 2 /2 - 3r/2 + 1 , and (c) First[Select[4r 3 /9 + 2r 2 /3 + {2, 5/3, 5/3} r - {10/9, 1, -4/9}, IntegerQ]] In any trivalent network, the quantity f[r] obtained by adding up the numbers of nodes reached by going distance r from each node must satisfy f[0] = n and f[1] = 3n , where n is the total number of nodes in the network. In addition, the limit of f[r] for large r must be n 2 .

Note (d) for Systems of Limited Size and Class 2 Behavior…Implementing boundary conditions [in cellular automata] In the bitwise representation discussed on page 865 , 0's outside of a width n can be implemented by applying BitAnd[a, 2 n -1] at each step.

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) . Assuming b > a > 0 , the number of zeros from the second family which appear between the n th and (n + 1) th zero from the first family is (Floor[(n + 1) #] - Floor[n #] &)[(b - a)/(a + b)] and as discussed on page 903 this sequence can be obtained by applying a sequence of substitution rules. For Sin[a x] + Sin[b x] a more complicated sequence of substitution rules yields the analogous sequence in which -1/2 is inserted in each Floor .

In case (a), the fraction of black elements fluctuates around 1/2; in (b) it approaches 3/4; in (d) it fluctuates around near 0.3548, while in (e) and (f) it does not appear to stabilize.

On the right-hand edge, the first few periods that are seen are {1, 2, 2, 4, 8, 8, 16, 32, 32, 64, 64, 64, 64, 64, 128, 256} and in general the period seems to increase exponentially with depth. On the left-hand edge, the period increases only extremely slowly: period 2 is first achieved at depth 3, period 4 at depth 8, 8 at 29, 16 at 400, 32 at 87,867, 64 at 2,107,985,255 or more, and so on. … The boundary that separates repetition on the left from randomness on the right moves an average of about 0.252 cells to the left at every step (compare page 949 ).

General powers [of numbers] It has been known in principle since the 1930s that Mod[h n , 1] is uniformly distributed in the range 0 to 1 for almost all values of h . … Exceptions are known to include so-called Pisot numbers such as GoldenRatio , √ 2 + 1 and Root[# 3 - # - 1 &, 1] (the numerically smallest of all Pisot numbers) for which Mod[h n , 1] becomes 0 or 1 for large n .

Properties [of number theoretic sequences] (a) The number of divisors of n is given by DivisorSigma[0, n] , equal to Length[Divisors[n]] . … It is known that the directions of all vectors {x, y, z} for which x 2 + y 2 + z 2  n are uniformly distributed in the limit of large n . … For d = 2 , this approaches π n for large n , with an error of order n c , where 1/4 < c ≤ 0.315 .

BesselJ[0, x] goes like Sin[x]/ √ x for large x while AiryAi[-x] goes like Sin[x 3/2 ]/x 1/4 . … (For AiryAi[x] the Stokes lines are in directions (-1)^({1, 2, 3}/3) .)

Symmetric 5-neighbor [2D cellular automaton] rules Among the 32 possible 5-cell neighborhoods shown for example on page 941 there are 12 classes related by symmetries, given by s = {{1}, {2, 3, 9, 17}, {4, 10, 19, 25}, {5}, {6, 7, 13, 21}, {8, 14, 23, 29}, {11, 18}, {12, 20, 26, 27}, {15, 22}, {16, 24, 30, 31}, {28}, {32}} Completely symmetric 5-neighbor rules can be numbered from 0 to 4095, with each digit specifying the new color of the cell for each of these symmetry classes of neighborhoods. Such rule numbers can be converted to general form using FromDigits[Map[Last, Sort[Flatten[Map[Thread, Thread[{s, IntegerDigits[n, 2, 12]}]], 1]]], 2]

Reversible mobile automata can for instance be constructed using Table[(IntegerDigits[i, 2, 3]  If[First[#]  0, {#, -1}, {Reverse[#], 1}]&)[IntegerDigits[perm 〚 i 〛 , 2, 3]], {i, 8}] where perm is an element of Permutations[Range[8]] . … Thus, for example, the system on page 121 based on successive multiplication by 3/2 is reversible by using division by 3/2.