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Python Program for Zeckendorf\’s Theorem (Non-Neighbouring Fibonacci Representation)

  • Last Updated : 07 Oct, 2021

Given a number, find a representation of number as sum of non-consecutive Fibonacci numbers.

Examples:

Input:  n = 10
Output: 8 2
8 and 2 are two non-consecutive Fibonacci Numbers
and sum of them is 10.

Input:  n = 30
Output: 21 8 1
21, 8 and 1 are non-consecutive Fibonacci Numbers
and sum of them is 30.

The idea is to use Greedy Algorithm

1) Let n be input number

2) While n >= 0
     a) Find the greatest Fibonacci Number smaller than n.
        Let this number be 'f'.  Print 'f'
     b) n = n - f 

Python




# Python program for Zeckendorf's theorem. It finds representation
# of n as sum of non-neighbouring Fibonacci Numbers.
 
# Returns the greatest Fibonacci Number smaller than
# or equal to n.
def nearestSmallerEqFib(n):
     
    # Corner cases
    if (n == 0 or n == 1):
        return n
        
    # Finds the greatest Fibonacci Number smaller
    # than n.
    f1, f2, f3 = 0, 1, 1
    while (f3 <= n):
        f1 = f2;
        f2 = f3;
        f3 = f1 + f2;
    return f2;
 
 
# Prints Fibonacci Representation of n using
# greedy algorithm
def printFibRepresntation(n):
     
    while (n>0):
 
        # Find the greates Fibonacci Number smaller
        # than or equal to n
        f = nearestSmallerEqFib(n);
  
        # Print the found fibonacci number
        print f,
  
        # Reduce n
        n = n-f
 
# Driver code test above functions
n = 30
print "Non-neighbouring Fibonacci Representation of", n, "is"
printFibRepresntation(n)
Output: 
Non-neighbouring Fibonacci Representation of 30 is
21 8 1

 

Please refer complete article on Zeckendorf’s Theorem (Non-Neighbouring Fibonacci Representation) for more details!
 




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