From the result on page 890 , the number whose digits are obtained from {1  {1, 0}, 0  {1}} is given by Sum[2^(-Floor[n GoldenRatio]), {n, ∞ }] .

Finding layouts [for networks] One way to lay out a network g so that network distances in it come as close as possible to ordinary distances in d -dimensional space, is just to search for values of the x[i, k] which minimize a quantity such as With[{n = Length[g]}, Apply[Plus, Flatten[(Table[Distance[g, {i, j}], {i, n}, {j, n}] 2 - Table[ Sum[(x[i, k] - x[j, k]) 2 , {k, d}], {i, n}, {j, n}]) 2 ]]] using for example FindMinimum starting say with x[1, _]  0 and all the other x[_, _]  Random[] .

Various schemes are used in practice, and almost all of them are based on the idea from traditional mathematics that by viewing data in terms of numbers it becomes possible to decompose the data into sums of fixed basic forms—some of which can be dropped in order to achieve compression.

Maximal block compression If one has data that consists of a long sequence of blocks, each of length b , and each independently chosen with probability p[i] to be of type i , then as argued by Claude Shannon in the late 1940s, it turns out that the minimum number of base 2 bits needed on average to represent each block in such a sequence is h = -Sum[p[i] Log[2, p[i]], {i, 2 b }] .

For initial conditions of size n , this occurs after at most Sum[Nest[2 # &, 0, i] - 1, {i, n}] + 1 steps.

A still better approximation is obtained by subtracting Sum[LogIntegral[n r i ], {i, - ∞ , ∞ }] where the r i are the complex zeros of the Riemann zeta function Zeta[s] , discussed on page 918 .

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 ).

One based on gradual extension of work by Richard Stoneham from 1971 is that numbers of the form Sum[1/(p n b p n ), {n, ∞ }] for prime p > 2 are normal in base b (for GCD[b, p]  1 ), and are transcendental.

MatrixPower[ m[Map[Length, list]], r] . w/Length[w]] then forming Sum[ ξ [Abs[r]] Cos[2 π r ω ], {r, -n/2, n/2}] and taking the limit n  ∞ .

The element at position n in the first sequence discussed above can however be obtained in about Log[n] steps using ((IntegerDigits[#3 + Quotient[#1, #2], 2] 〚 Mod[#1, #2] + 1 〛 &)[n - (# - 2)2 # - 1 - 2, #, 2 # - 1 ]&)[NestWhile[# + 1&, 0, (# - 1)2 # + 1 < n &]] where the result of the NestWhile can be expressed as Ceiling[1 + ProductLog[1/2(n - 1)Log[2]]/Log[2]] Following work by Maxim Rytin in the late 1990s about k n+1 digits of a concatenation sequence can be found fairly efficiently from k/(k - 1) 2 - (k - 1) Sum[k (k s - 1) ((1 + s - s k)/(k - 1)) (1/((k - 1) (k s - 1) 2 ) - k/((k - 1) (k s + 1 - 1) 2 ) + 1/(k s + 1 - 1)), {s, n}] Concatenation sequences can also be generated by joining together digits from other representations of numbers; the picture below shows results for the Gray code representation from page 901 .