In the first 200 billion digits, the frequencies of 0 through 9 differ from 20 billion by {30841, -85289, 136978, 69393, -78309, -82947, -118485, -32406, 291044, -130820} An early approximation to π was 4 Sum[(-1) k /(2k + 1), {k, 0, m}] 30 digits were obtained with 2 Apply[Times, 2/Rest[NestList[Sqrt[2 + #]&, 0, m]]] An efficient way to compute π to n digits of precision is (# 〚 2 〛 2 /# 〚 3 〛 )& [NestWhile[Apply[Function[{a, b, c, d}, {(a + b)/2, Sqrt[a b], c - d (a - b) 2 , 2 d}], #]&, {1, 1/Sqrt[N[2, n]], 1/4, 1/4}, # 〚 2 〛 ≠ # 〚 2 〛 &]] This requires about Log[2, n] steps, or a total of roughly n Log[n] 2 operations (see page 1134 ).

The possible sequences of length n blocks that can occur are conveniently represented by possible paths by so-called de Bruijn networks, of the kind shown for k = 2 and n = 2 through 5 below.

Geological structures Typical landscapes on Earth are to a first approximation formed by regions of crust being uplifted through tectonic activity, then being sculpted by progressive erosion (and redeposition of sediment) associated with the flow of water.

Note that while many of the people listed are well known, extensive research (often through personal contacts, as well as institutional and government records) was required to track down quite a few of them.

But such a block corresponds in a sense to a horizontal slice through the cone of previous cell colors.

Examples are early mechanical automata and model airplanes, and modern executive toys claiming to illustrate chaos theory through linkages, magnets or fluid systems.

Biological systems contain many specific alkanes—often quite large—that happen to be produced through chemical pathways in biological cells.

As a increases, the repetition period goes through a series of doublings.

And a large part of the way this has been done in practice is just through comparison with known examples that fit into an overall pattern of gradual historical change.

The alternative pragmatic theory popular in rational philosophy and economics suggests that behavior in legal matters is determined through calculations based on laws and the deterrents they provide.