(Typically one needs to generalize formulas that are initially set up with integer numbers of terms; examples include taking Power[x, y] to be Exp[Log[x] y] and x!

The number of sequences s n of length n that can actually occur is given by Apply[Plus, Flatten[MatrixPower[m, n]]] where the adjacency matrix m is given by MapAt[(1 + #) &, Table[0, {Length[net]}, {Length[net]}], Flatten[MapIndexed[{First[#2], Last[#1]} &, net, {2}], 1]] For rule 32, for example, s n turns out to be Fibonacci[n + 3] , so that for large n it is approximately GoldenRatio n .

Unlike for continuous mathematical functions, known algorithms for number theoretical functions such as FactorInteger[x] or MoebiusMu[x] typically seem to require a number of operations that grows faster with the number of digits n in x than any power of n (see page 1090 ).

A typical collection of tests described by Donald Knuth in 1968 includes: (1) frequency or equidistribution test (possible elements should occur with equal frequency); (2) serial test (pairs of elements should be equally likely to be in descending and ascending order); (3) gap test (runs of elements all greater or less than some fixed value should have lengths that follow a binomial distribution); (4) poker test (blocks corresponding to possible poker hands should occur with appropriate frequencies); (5) coupon collector's test (runs before complete sets of values are found should have lengths that follow a definite distribution); (6) permutation test (in blocks of elements possible orderings of values should occur equally often); (7) runs up test (runs of monotonically increasing elements should have lengths that follow a definite distribution); (8) maximum-of-t test (maximum values in blocks of elements should follow a power-law distribution).

And just as with fractal dimensions discussed on page 933 there are issues about whether a definite power law for the growth rate will emerge, and how one should average over results for different parts of the network.

Given the functional equation one can find a power series for g[x] for any a .

The detection distance increases like the square root of the signal strength, covering all 10 11 stars in our galaxy when the signal uses the total power output of a star.

Turing and Post seem to have thought of Church's Thesis as characterizing the "mathematicizing power" of humans, and Turing at least seems to have thought that it might not apply to continuous processes in physics.

How this could be consistent with God having infinite power was not clear, although around 420 AD Augustine suggested that while God might have infinite knowledge of the future we as humans could not—yielding what can be viewed as a very rough analog of my explanation for free will.

Fibonacci[n] can be obtained in many ways: • (GoldenRatio n - (-GoldenRatio) -n )/ √ 5 • Round[GoldenRatio n / √ 5 ] • 2 1 - n Coefficient[(1 + √ 5 ) n , √ 5 ] • MatrixPower[{{1, 1}, {1, 0}}, n - 1] 〚 1, 1 〛 • Numerator[NestList[1/(1 + #)&, 1, n]] • Coefficient[Series[1/(1 - t - t 2 ), {t, 0, n}], t n - 1 ] • Sum[Binomial[n - i - 1, i], {i, 0, (n - 1)/2}] • 2 n - 2 - Count[IntegerDigits[Range[0, 2 n - 2 ], 2], {___, 1, 1, ___}] A fast method for evaluating Fibonacci[n] is First[Fold[f, {1, 0, -1}, Rest[IntegerDigits[n, 2]]]] f[{a_, b_, s_}, 0] = {a (a + 2b), s + a (2a - b), 1} f[{a_, b_, s_}, 1] = {-s + (a + b) (a + 2b), a (a + 2b), -1} Fibonacci numbers appear to have first arisen in perhaps 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths.