Lucas numbers Lucas numbers Lucas[n] satisfy the same recurrence relation f[n_] := f[n - 1] + f[n - 2] as Fibonacci numbers, but with the initial conditions f[1] = 1 ; f[2] = 3 . Among the relations satisfied by Lucas numbers are: • Lucas[n_] := Fibonacci[n - 1] + Fibonacci[n + 1] • GoldenRatio n  (Lucas[n] + Fibonacci[n] √ 5 )/2

Successive approximations to this number are given by Fibonacci[n - 2]/Fibonacci[n] , so that elements numbered Fibonacci[n] (i.e. 1, 2, 3, 5, 8, 13, ...) will be the ones that come closest to being a whole number of turns apart, and thus to being lined up on the stem.

Fibonacci numbers The Fibonacci numbers Fibonacci[n] ( f[n] for short) can be generated by the recurrence relation f[n_] := f[n] = f[n-1] + f[n - 2] f[1] = f[2] = 1 The first few Fibonacci numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. … For m > 1 , the value of n for which m  Fibonacci[n] is Round[Log[GoldenRatio, √ 5 m]] . … Mod[Fibonacci[n], n] has the fairly complicated form shown below.

Generalized Fibonacci sequences Any linear recurrence relation yields sequences with many properties in common with the Fibonacci numbers—though with GoldenRatio replaced by other algebraic numbers.

For the Fibonacci encoding used in the main text, this factor is approximately 1.41028. (In base 2 this number has 1's essentially at positions Fibonacci[n] ; as discussed on page 914 , the number is transcendental.)

The pictures below show the results with k = 2 (rule 60) for (a) Fibonacci[n] , (b) Thue–Morse sequence, (c) Fibonacci substitution system, (d) (Prime[n] - 1)/2 , (e) digits of π .

And for example from the fact that x 2  y 2 + (x y ± 1) has solutions Fibonacci[n] it follows that the positive values of (2 - (x 2 - y 2 - x y) 2 )x are just Fibonacci[n] (achieved when {x, y} is Fibonacci[{n, n - 1}] ). This is the simplest polynomial giving Fibonacci[n] , and there are for example no polynomials with 2 variables, up to 4 terms, total degree less than 4, and integer coefficients between -2 and +2, that give any of 2 n , 3 n or Prime[n] .

Flatten[IntegerDigits[ Append[2 - With[{w = Floor[Log[3, 2n]]}, IntegerDigits[n - (3 w + 1 - 1)/2, 3, w]], 3], 2, 2]] (e) Fibonacci encoding. Instead of decomposing a number into a sum of powers of an integer base, one decomposes it into a sum of Fibonacci numbers (see page 902 ). … Apply[Take, RealDigits[(N[#, N[Log[10, #] + 3]] &)[ n √ 5 /GoldenRatio 2 + 1/2], GoldenRatio]] The representations of all the first Fibonacci[n] - 1 numbers can be obtained from (the version in the main text has Rest[RotateLeft[Join[#, {0, 1}]]] & applied) Apply[Join, Map[Last, NestList[{# 〚 2 〛 ], Join[Map[Join[{1, 0}, Rest[#]] & , # 〚 2 〛 ], Map[Join[{1, 0}, #] &, # 〚 1 〛 ]]} &, {{}, {{1}}}, n-3]]]

Other uniformly distributed sequences Cases in which Mod[a[n], 1] is uniformly distributed include √ n , n Log[n] , Log[Fibonacci[n]] , Log[n!]

, 2 n or Fibonacci[n] .