# 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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