Collection of Mathematical Formulas/Fibonacci Numbers

      Definition of the Fibonacci sequence:

      \begin{cases} F_0 = 0 \\ F_1 = 1 \\ F_{n+2} = F_n+F_{n+1}, &n = 0, 1, 2, 3, 5, 8 ... \end{cases} (1)

      [TAOCP1, p. 83], [Concrete, p. 299]:

      F_n = \frac{1}{\sqrt{5}} (\phi^n - \hat{\phi}^n), \quad n \in \mathbb{Z}, \quad \phi = \frac{1+\sqrt{5}}{2}, \hat{\phi} = \frac{1-\sqrt{5}}{2} (2)

      [TAOCP1, p. 85]:

      F_{n+k} F_{m-k} - F_n F_m = (-1)^n F_{m-n-k} F_k (2)
      Last modified on 24 October 2011, at 11:37