For the Fibonacci sequence, it is instead n - 1 . The maximum number of distinct nodes at any level in the tree has large fluctuations but its peaks seem to increase roughly linearly for all the rules on this page (in the Fibonacci case it is Ceiling[n/2] ).

For it turns out that an angle between successive elements of about 137.5° is equivalent to a rotation by a number of turns equal to the so-called golden ratio (1+Sqrt[5])/2 ≃ 1.618 which arises in a wide variety of mathematical contexts—notably as the limiting ratio of Fibonacci numbers.

The n th term in its continued fraction representation turns out to be 2^Fibonacci[n - 2] .

One approach is to use {1, 1} to indicate the boundary of each block, and then within each block to use all possible digit sequences which do not contain {1, 1} , as in the Fibonacci number system discussed on page 892 .

The Hindu-Arabic base 10 system in its modern form probably originated around 600 AD, and particularly following the work of Leonardo Fibonacci in the early 1200s, became common by the 1400s.

(b) (Fibonacci-related sequence) This sequence is a Sturmian one. The maximum of the spectrum is at Fibonacci[t] .

The arrangement of triangles at step t can be obtained from a substitution system according to With[{ ϕ = GoldenRatio}, Nest[# /. a[p_, q_, r_]  With[{s = (p + ϕ q) (2 - ϕ )}, {a[r, s, q], b[r, s, p]}] /. b[p_, q_, r_]  With[{s = (p + ϕ r) (2 - ϕ )}, {a[p, q, s], b[ r, s, q]}] &, a[{1/2, Sin[2 π /5] ϕ }, {1, 0}, {0, 0}], t]] This pattern can be viewed as generalizations of the pattern generated by the 1D Fibonacci substitution system (c) on page 83 .

Of the 4 billion k = 2 , r = 2 cellular automata none turn out to be able to produce for example sequences corresponding to the cubes, powers of 3, Fibonacci numbers, primes, digits of √ 2 , or concatenation sequences.

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 .

But given a sequence of digits that are each 0 or 1, it is also possible for example to construct numbers according to Sum[a[i] Fibonacci[i + 2], {i, 0, r}] .