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