Most arose first as solutions to specific differential equations, typically in physics and astronomy; some arose as products, sums of series or inverses of other functions.

If one associates with each possible sequence of length n a number Sum[a i 2 -i , {i, n}] , then the set of sequences that actually occur at a given step form a Cantor set (see note below ), whose Hausdorff dimension turns out to be exactly h .

The overall fraction of a length n input that consists of repeats of length at least b is greater than 1 - 2 b /n and is essentially 1 - Sum[(1 - 2 -b ) i Product[1 + (1 - 2 -b ) j - (1 - 2 -b - 1 ) j , {j, i - b + 1, i - 1}], {i, b, n - b}]/(n - 2b + 1)

Some integer functions can readily be obtained by supplying integer arguments to continuous functions, so that for example Mod[x, 2] corresponds to Sin[ π x/2] 2 or (1 - Cos[ π x])/2, Mod[x, 3] ↔ 1 + 2/3(Cos[2/3 π (x - 2)] - Cos[2 π x/3]) Mod[x, 4] ↔ (3 - 2 Cos[ π x/2] - Cos[ π x] - 2 Sin[ π x/2])/2 Mod[x, n] ↔ Sum[j Product[(Sin[ π (x - i - j)/n]/ Sin[ π i/n]) 2 , {i, n - 1}], {j, n - 1}] (As another example, If[x > 0, 1, 0] corresponds to 1 - 1/Gamma[1 - x] .)

In 1912 Karl Sundman did however find an infinite series that could in principle be summed to give the solution—but which converges exceptionally slowly.

(Other appropriate primitives may conceivably be related to the solubility of Hilbert's Thirteenth Problem and the fact that any continuous function with any number of arguments can be written as a one-argument function of a sum of a handful of fixed one-argument functions applied to the arguments of the original function.)

(The evaluation is based on the Riemann–Siegel formula, which involves sums of about √ x cosines.)

And these amplitudes a i are assumed to be complex numbers with a continuous range of possible values, subject only to the conventional constraint of unit total probability Sum[Abs[a i ] 2 , {i, 2 n }]  1 . … In a classical system like a cellular automaton with n cells a probabilistic ensemble of states can similarly be described by a vector of 2 n probabilities p i —now satisfying Sum[p i , {i, 2 n }]  1 , and evolving by multiplication with 2 n × 2 n matrices having a single 1 in each row.

As discussed in the note below, this can be viewed as a consequence of the fact that the probability distribution in a random walk depends only on Sum[Outer[Times, e 〚 s 〛 , e 〚 s 〛 ], {s, Length[e]}] and not on products of more of the e 〚 s 〛 .

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.