Python Program for Fibonacci numbers

The Fibonacci numbers are the numbers in the following integer sequence.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ……..
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation 

    Fn = Fn-1 + Fn-2

with seed values  

   F0 = 0 and F1 = 1.

Method 1 ( Use recursion ) :

Python3

# Function for nth Fibonacci number
def Fibonacci(n):

    # Check if input is 0 then it will
    # print incorrect input
    if n < 0:
        print("Incorrect input")

    # Check if n is 0
    # then it will return 0
    elif n == 0:
        return 0

    # Check if n is 1,2
    # it will return 1
    elif n == 1 or n == 2:
        return 1

    else:
        return Fibonacci(n-1) + Fibonacci(n-2)

# Driver Program
print(Fibonacci(9))

# This code is contributed by Saket Modi
# then corrected and improved by Himanshu Kanojiya

Output

34

Time complexity: O(2 ^ n)  Exponential
Auxiliary Space: O(n)

Method 2 ( Use Dynamic Programming ) :

Python3

# Function for nth fibonacci
# number - Dynamic Programming
# Taking 1st two fibonacci numbers as 0 and 1
FibArray = [0, 1]

def fibonacci(n):

    # Check is n is less
    # than 0
    if n < 0:
        print("Incorrect input")

    # Check is n is less
    # than len(FibArray)
    elif n < len(FibArray):
        return FibArray[n]
    else:
        FibArray.append(fibonacci(n - 1) + fibonacci(n - 2))
        return FibArray[n]

# Driver Program
print(fibonacci(9))

# This code is contributed by Saket Modi

Output

34

Time complexity: O(n)
Auxiliary Space: O(n)

Method 3 ( Space Optimized):

Python

# Function for nth fibonacci
# number - Space Optimisation
# Taking 1st two fibonacci numbers as 0 and 1

def fibonacci(n):
    a = 0
    b = 1

    # Check is n is less
    # than 0
    if n < 0:
        print("Incorrect input")

    # Check is n is equal
    # to 0
    elif n == 0:
        return 0

    # Check if n is equal to 1
    elif n == 1:
        return b
    else:
        for i in range(1, n):
            c = a + b
            a = b
            b = c
        return b

# Driver Program
print(fibonacci(9))

# This code is contributed by Saket Modi
# Then corrected and improved by Himanshu Kanojiya

Output

34

Time complexity: O(n)
Auxiliary Space: O(1)

Method 4 (Cache):  

Python3

from functools import lru_cache

# Function for nth Fibonacci number

# lru_cache will store the result
# so we don't have to find
# fibonacci for same num again


@lru_cache(None)
def fibonacci(num: int) -> int:

    # check if num is less than 0
    # it will return none
    if num < 0:
        print("Incorrect input")
        return

    # check if num between 1, 0
    # it will return num
    elif num < 2:
        return num

    # return the fibonacci of num - 1 & num - 2
    return fibonacci(num - 1) + fibonacci(num - 2)


# Driver Program
print(fibonacci(9))

# This code is contributed by Sayedul Haque Sarker

Output

34

Time complexity: O(n)
Auxiliary Space: O(n)

Method 5(Using Backtracking):

Python3

def fibonacci(n, memo={}):
    if n <= 0:
        return 0
    elif n == 1:
        return 1
    elif n in memo:
        return memo[n]
    else:
        memo[n] = fibonacci(n-1) + fibonacci(n-2)
        return memo[n]

# Driver Program
print(fibonacci(9))

Output

34

Time complexity: O(n)
Auxiliary Space: O(n)

Please refer complete article on Program for Fibonacci numbers for more details!