[7] Offer to prove safety Fibonacci columns python achieve

Title Description

We all know that Fibonacci number, and now asked to enter an integer n, you output the n-th Fibonacci number Fibonacci sequence of item (from 0, the first 0 is 0). n <= 39

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1. recursion

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1. Analysis
   Fibonacci standard formula of Number is: F (1) = 1, F (2) = 1, F (n) = F (n-1) + (n> = 3 F (n-2), n ∈N *)
   according to the formula directly write:

2. Code

3. Complexity
   Time Complexity: $O (2 ^ n) O (2 n)$
   spatial complexity: $O (1) O (1)$

2. Optimize recursive

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1. Analysis of
   recursive calculation is repeated a lot of the same data, we use an array to store the result up 8!

2. Code

3. Complexity
   time complexity: O (n) O (n )
   space complexity: O (n) O (n )

3. Optimizing Storage

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1. Analysis of
   fact, we can find a time to use the two most recent numbers, so we can store only the last two numbers

• sum of n items stored value
• one stored value of the n-1 items
• the stored value of the two items n-2

2. Code

# -*- coding:utf-8 -*-
class Solution:
    def Fibonacci(self, n):
        # write code here
        preNum = 1
        prePreNum = 0

        result = 0
        if n == 0:
            return 0
        if n == 1:
            return 1
        # [0,1,1,2]
        for i in range(2,n+1):

            result = preNum + prePreNum
            prePreNum = preNum
            preNum = result

        return result

3. Complexity:
   time complexity: O (n) O (n )
   complexity of space: O (1) O (1 )