Editing FibonacciSequence
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The Fibonacci Sequence is obtained by starting with 0 and 1, then getting each successive term by adding the two previous terms. * /F(0)=0/ * /F(1)=1/ * /F(n)=F(n-1)+F(n-2)/ This gives: * 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... Ratios of successive terms approach the golden ratio. [[[>50 The closed form of the Fibonacci Sequence can be derived by using matrices and computing the eigenvalues. That's because we have the formula |>> EQN:\left[\begin{matrix}F_n\\F_{n+1}\end{matrix}\right]=\left[\begin{matrix}0&1\\1&1\end{matrix}\right]\left[\begin{matrix}F_{n-1}\\F_n\end{matrix}\right] <<| That means |>> EQN:\left[\begin{matrix}F_n\\F_{n+1}\end{matrix}\right]=\left[\begin{matrix}0&1\\1&1\end{matrix}\right]^n\left[\begin{matrix}0\\1\end{matrix}\right] <<| and so if only we could compute powers rapidly and easily, then we could compute the Fibonacci Sequence. But that's what eigenvectors and eigenvalues do for us. ]]] Closed form: EQN:F(n)=\frac{\phi^n-(-\phi)^{-n}}{\sqrt{5}} where EQN:\phi is the golden ratio. Since EQN:\phi^{\small-n} approaches 0, we can ignore that term and say that /F(n)/ is the closest integer to EQN:\phi^n/\sqrt{5} The Fibonacci sequence turns up in all sorts of places in nature: * Number of spirals on a ** pine cone ** pineapple ** sunflower centre etc. Compare with the Perrin Sequence. ---- * http://en.wikipedia.org/wiki/Fibonacci_number * http://mathworld.wolfram.com/FibonacciNumber.html * http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html * http://www.google.com/search?q=fibonacci+sequence