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