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