# Ways to represent a number as a sum of 1’s and 2’s

Given a positive integer N. The task is to find the number of ways of representing N as a sum of 1s and 2s.

Examples:

```Input : N = 3
Output : 3
3 can be represented as (1+1+1), (2+1), (1+2).

Input : N = 5
Output : 8
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

For N = 1, answer is 1.
For N = 2. (1 + 1), (2), answer is 2.
For N = 3. (1 + 1 + 1), (2 + 1), (1 + 2), answer is 3.
For N = 4. (1 + 1 + 1 + 1), (2 + 1 + 1), (1 + 2 + 1), (1 + 1 + 2), (2 + 2) answer is 5.
And so on.

It can be observe that it form Fibonacci Series. So, the number of ways of representing N as a sum of 1s and 2s is (N + 1)th Fibonacci number.
How ?
We can easily see that the recursive function is exactly same as Fibonacci Numbers. To obtain the sum of N, we can add 1 to N – 1. Also, we can add 2 to N – 2. And only 1 and 2 are allowed to make the sum N. So, to obtain sum N using 1s and 2s, total ways are: number of ways to obtain (N – 1) + number of ways to obtain (N – 2).

We can find N’th Fibonacci Number in O(Log n) time. Please refer method 5 of this post.

Below is the implementation of this approach:

## C++

 `// C++ program to find number of ways to representing ` `// a number as a sum of 1's and 2's ` `#include ` `using` `namespace` `std; ` ` `  `// Function to multiply matrix. ` `void` `multiply(``int` `F[2][2], ``int` `M[2][2]) ` `{ ` `    ``int` `x =  F[0][0]*M[0][0] + F[0][1]*M[1][0]; ` `    ``int` `y =  F[0][0]*M[0][1] + F[0][1]*M[1][1]; ` `    ``int` `z =  F[1][0]*M[0][0] + F[1][1]*M[1][0]; ` `    ``int` `w =  F[1][0]*M[0][1] + F[1][1]*M[1][1]; ` ` `  `    ``F[0][0] = x; ` `    ``F[0][1] = y; ` `    ``F[1][0] = z; ` `    ``F[1][1] = w; ` `} ` ` `  `// Power function in log n ` `void` `power(``int` `F[2][2], ``int` `n) ` `{ ` `    ``if``( n == 0 || n == 1) ` `        ``return``; ` `    ``int` `M[2][2] = {{1,1},{1,0}}; ` ` `  `    ``power(F, n/2); ` `    ``multiply(F, F); ` ` `  `    ``if` `(n%2 != 0) ` `        ``multiply(F, M); ` `} ` ` `  `/* function that returns (n+1)th Fibonacci number ` `   ``Or number of ways to represent n as sum of 1's ` `   ``2's */` `int` `countWays(``int` `n) ` `{ ` `    ``int` `F[2][2] = {{1,1},{1,0}}; ` `    ``if` `(n == 0) ` `        ``return` `0; ` `    ``power(F, n); ` `    ``return` `F[0][0]; ` `} ` ` `  `// Driver program ` `int` `main() ` `{ ` `    ``int` `n = 5; ` `    ``cout << countWays(n) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java program to find number of  ` `// ways to representing a number ` `// as a sum of 1's and 2's ` `class` `GFG  ` `{ ` ` `  `// Function to multiply matrix. ` `    ``static` `void` `multiply(``int` `F[][], ``int` `M[][]) ` `    ``{ ` `        ``int` `x = F[``0``][``0``] * M[``0``][``0``] + F[``0``][``1``] * M[``1``][``0``]; ` `        ``int` `y = F[``0``][``0``] * M[``0``][``1``] + F[``0``][``1``] * M[``1``][``1``]; ` `        ``int` `z = F[``1``][``0``] * M[``0``][``0``] + F[``1``][``1``] * M[``1``][``0``]; ` `        ``int` `w = F[``1``][``0``] * M[``0``][``1``] + F[``1``][``1``] * M[``1``][``1``]; ` ` `  `        ``F[``0``][``0``] = x; ` `        ``F[``0``][``1``] = y; ` `        ``F[``1``][``0``] = z; ` `        ``F[``1``][``1``] = w; ` `    ``} ` ` `  `    ``// Power function in log n ` `    ``static` `void` `power(``int` `F[][], ``int` `n)  ` `    ``{ ` `        ``if` `(n == ``0` `|| n == ``1``)  ` `        ``{ ` `            ``return``; ` `        ``} ` `        ``int` `M[][] = {{``1``, ``1``}, {``1``, ``0``}}; ` ` `  `        ``power(F, n / ``2``); ` `        ``multiply(F, F); ` ` `  `        ``if` `(n % ``2` `!= ``0``)  ` `        ``{ ` `            ``multiply(F, M); ` `        ``} ` `    ``} ` ` `  `    ``/* function that returns (n+1)th Fibonacci number ` `    ``Or number of ways to represent n as sum of 1's ` `    ``2's */` `    ``static` `int` `countWays(``int` `n)  ` `    ``{ ` `        ``int` `F[][] = {{``1``, ``1``}, {``1``, ``0``}}; ` `        ``if` `(n == ``0``) ` `        ``{ ` `            ``return` `0``; ` `        ``} ` `        ``power(F, n); ` `        ``return` `F[``0``][``0``]; ` `    ``} ` ` `  `    ``// Driver program ` `    ``public` `static` `void` `main(String[] args)  ` `    ``{ ` `        ``int` `n = ``5``; ` `        ``System.out.println(countWays(n)); ` `    ``} ` `} ` ` `  `// This code contributed by Rajput-Ji `

## Python3

 `# Python3 program to find number of ways to  ` `# representing a number as a sum of 1's and 2's ` ` `  `# Function to multiply matrix. ` `def` `multiply(F, M): ` ` `  `    ``x ``=` `F[``0``][``0``] ``*` `M[``0``][``0``] ``+` `F[``0``][``1``] ``*` `M[``1``][``0``] ` `    ``y ``=` `F[``0``][``0``] ``*` `M[``0``][``1``] ``+` `F[``0``][``1``] ``*` `M[``1``][``1``] ` `    ``z ``=` `F[``1``][``0``] ``*` `M[``0``][``0``] ``+` `F[``1``][``1``] ``*` `M[``1``][``0``] ` `    ``w ``=` `F[``1``][``0``] ``*` `M[``0``][``1``] ``+` `F[``1``][``1``] ``*` `M[``1``][``1``] ` ` `  `    ``F[``0``][``0``] ``=` `x ` `    ``F[``0``][``1``] ``=` `y ` `    ``F[``1``][``0``] ``=` `z ` `    ``F[``1``][``1``] ``=` `w ` ` `  `# Power function in log n ` `def` `power(F, n): ` ` `  `    ``if``( n ``=``=` `0` `or` `n ``=``=` `1``): ` `        ``return` `    ``M ``=` `[[``1``, ``1``],[``1``, ``0``]] ` ` `  `    ``power(F, n ``/``/` `2``) ` `    ``multiply(F, F) ` ` `  `    ``if` `(n ``%` `2` `!``=` `0``): ` `        ``multiply(F, M) ` ` `  `#/* function that returns (n+1)th Fibonacci number ` `# Or number of ways to represent n as sum of 1's ` `# 2's */ ` `def` `countWays(n): ` `    ``F ``=` `[[``1``, ``1``], [``1``, ``0``]] ` `    ``if` `(n ``=``=` `0``): ` `        ``return` `0` `    ``power(F, n) ` ` `  `    ``return` `F[``0``][``0``] ` ` `  `# Driver Code ` `n ``=` `5` `print``(countWays(n)) ` ` `  `# This code is contributed by mohit kumar `

## C#

 `// C# program to find number of  ` `// ways to representing a number ` `// as a sum of 1's and 2's ` `class` `GFG  ` `{ ` ` `  `    ``// Function to multiply matrix. ` `    ``static` `void` `multiply(``int` `[,]F, ``int` `[,]M) ` `    ``{ ` `        ``int` `x = F[0,0] * M[0,0] + F[0,1] * M[1,0]; ` `        ``int` `y = F[0,0] * M[0,1] + F[0,1] * M[1,1]; ` `        ``int` `z = F[1,0] * M[0,0] + F[1,1] * M[1,0]; ` `        ``int` `w = F[1,0] * M[0,1] + F[1,1] * M[1,1]; ` ` `  `        ``F[0,0] = x; ` `        ``F[0,1] = y; ` `        ``F[1,0] = z; ` `        ``F[1,1] = w; ` `    ``} ` ` `  `    ``// Power function in log n ` `    ``static` `void` `power(``int` `[,]F, ``int` `n)  ` `    ``{ ` `        ``if` `(n == 0 || n == 1)  ` `        ``{ ` `            ``return``; ` `        ``} ` `        ``int` `[,]M = {{1, 1}, {1, 0}}; ` ` `  `        ``power(F, n / 2); ` `        ``multiply(F, F); ` ` `  `        ``if` `(n % 2 != 0)  ` `        ``{ ` `            ``multiply(F, M); ` `        ``} ` `    ``} ` ` `  `    ``/* function that returns (n+1)th Fibonacci number ` `    ``Or number of ways to represent n as sum of 1's ` `    ``2's */` `    ``static` `int` `countWays(``int` `n)  ` `    ``{ ` `        ``int` `[,]F = {{1, 1}, {1, 0}}; ` `        ``if` `(n == 0) ` `        ``{ ` `            ``return` `0; ` `        ``} ` `        ``power(F, n); ` `        ``return` `F[0,0]; ` `    ``} ` ` `  `    ``// Driver program ` `    ``public` `static` `void` `Main()  ` `    ``{ ` `        ``int` `n = 5; ` `        ``System.Console.WriteLine(countWays(n)); ` `    ``} ` `} ` ` `  `// This code contributed by mits `

Output:

```8
```

Time Complexity: O(logn).

This article is contributed by Anuj Chauhan. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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