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

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 program for Zeckendorf's theorem. It finds representation ` `# of n as sum of non-neighbouring Fibonacci Numbers. ` ` `  `# Returns the greatest Fibonacci Numberr 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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