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 algorithmdef 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 functionsn = 30print "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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