# 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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