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

• Last Updated : 23 Jun, 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 ```

## CPP

 `// C++ program for Zeckendorf's theorem. It finds representation` `// of n as sum of non-neighbouring Fibonacci Numbers.` `#include ` `using` `namespace` `std;`   `// Returns the greatest Fibonacci Number smaller than` `// or equal to n.` `int` `nearestSmallerEqFib(``int` `n)` `{` `    ``// Corner cases` `    ``if` `(n == 0 || n == 1)` `        ``return` `n;`   `    ``// Find the greatest Fibonacci Number smaller` `    ``// than n.` `    ``int` `f1 = 0, f2 = 1, f3 = 1;` `    ``while` `(f3 <= n) {` `        ``f1 = f2;` `        ``f2 = f3;` `        ``f3 = f1 + f2;` `    ``}` `    ``return` `f2;` `}`   `// Prints Fibonacci Representation of n using` `// greedy algorithm` `void` `printFibRepresntation(``int` `n)` `{` `    ``while` `(n > 0) {` `        ``// Find the greates Fibonacci Number smaller` `        ``// than or equal to n` `        ``int` `f = nearestSmallerEqFib(n);`   `        ``// Print the found fibonacci number` `        ``cout << f << ``" "``;`   `        ``// Reduce n` `        ``n = n - f;` `    ``}` `}`   `// Driver method to test` `int` `main()` `{` `    ``int` `n = 30;` `    ``cout << ``"Non-neighbouring Fibonacci Representation of "` `        ``<< n << ``" is \n"``;` `    ``printFibRepresntation(n);` `    ``return` `0;` `}`

Output:

```Non-neighbouring Fibonacci Representation of 30 is
21 8 1```

Time Complexity: O(n)

Auxiliary Space: O(1)

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

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