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

Last Updated : 20 Jan, 2022

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 ```

## Python3

 `# 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,end``=``" "``)` ` `  `        ``# 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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