Notes

Chapter 3: The World of Simple Programs

Section 5: Substitution Systems


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] Sqrt[5])/2


From Stephen Wolfram: A New Kind of Science [citation]