This is achieved for example by the definitions f[n_] := f[n] = f[n - f[n - 1]] + f[n - f[n - 2]] f[1] = f[2] = 1 The question of which recursive definitions yield meaningful sequences can depend on the details of how the rules are applied. … The default form of evaluation for recursive functions implemented by all standard computer languages (including Mathematica) is the so-called leftmost innermost scheme, which attempts to find explicit values for each f[k] that occurs first, and will therefore never notice if f[k] in fact occurs only in the combination f[k] - f[k] .

Another approach to computing square roots is based on the fact that the ratio of successive terms in for example the sequence f[i] = 2 f[i - 1] + f[i - 2] with f[1] = f[2] = 1 tends to 1 + √ 2 . This method yields about 2.5 t base 2 digits after t steps. … Note that the method works not only for integers, but for any rational number n for which 1 ≤ n < 4 .

They may argue that it is only robust and invariant concepts that are useful.

But inevitably the fit will only be approximate, so there will always be room for effects from randomness in the environment.

And the Black–Scholes model from 1973 implies that prices of suitably constructed options should depend in a sense only on such variances.

But its predictions like E = m c 2 were immediately applied for example to radioactivity, and soon it came to be assumed that the theory would work for any system at all—unless it involved gravity. So this has meant that in particle physics c 2 t 2 - x 2 - y 2 - z 2 is at some level the only quantity that ever appears.

For in making a summary one inevitably has to pick out only certain features, and in doing this one can remove or obscure the most interesting effects.

The infinite repetitive pattern shown here, together with its rotations and reflections, is the only one that satisfies this constraint.

And the crucial point is that I believe that this universality extends not only across simple programs, but also to systems in nature.

If we had found only a few examples of programs that could generate randomness in this way, then we might think that this third mechanism was a rare and special one.