Given an positive even integer ‘n’. Count total number of ways to express ‘n’ as sum of even positive integers. Output the answer in modulo **10 ^{9} + 7**

**Examples:**

Input: 6 Output: 4ExplanationThere are only four ways to write 6 as sum of even integers: 1. 2 + 2 + 2 2. 2 + 4 3. 4 + 2 4. 6 Input: 8 Output: 8

**Approach** is to find pattern or recursive function whichever is possible. The approach would be the same as already discussed in “Count ways to express ‘n’ as sum of odd integers“. Here the given number is even that means even sum can only be achieved by adding the (n-2)^{th} number as two times. We can notice that (by taking some examples) adding a 2 to a number doubles the count. Let the total number of ways to write ‘n’ be ways(n). The value of ‘ways(n)’ can be written by formula as follows:

ways(n) = ways(n-2) + ways(n-2) ways(n) = 2 * ways(n-2) ways(2) = 1 = 2^{0}ways(4) = 2 = 2^{1}ways(6) = 4 = 2^{2}ways(8) = 8 = 2^{3}'' '' '' ways(2 * n) = 2^{n-1}Replace n by (m / 2) =>ways(m) = 2^{m/2 - 1}

## C++

`// C++ program to count ways to write ` `// number as sum of even integers ` `#include<iostream> ` `using` `namespace` `std; ` ` ` `// Initialize mod variable as constant ` `const` `int` `MOD = 1e9 + 7; ` ` ` `/* Iterative Function to calculate (x^y)%p in O(log y) */` `int` `power(` `int` `x, unsigned ` `int` `y, ` `int` `p) ` `{ ` ` ` `int` `res = 1; ` `// Initialize result ` ` ` ` ` `x = x % p; ` `// Update x if it is more than or ` ` ` `// equal to p ` ` ` ` ` `while` `(y > 0) ` ` ` `{ ` ` ` `// If y is odd, multiply x with result ` ` ` `if` `(y & 1) ` ` ` `res = (1LL * res * x) % p; ` ` ` ` ` `// y must be even now ` ` ` `y = y>>1; ` `// y = y/2 ` ` ` `x = (1LL * x * x) % p; ` ` ` `} ` ` ` `return` `res; ` `} ` ` ` `// Return number of ways to write 'n' ` `// as sum of even integers ` `int` `countEvenWays(` `int` `n) ` `{ ` ` ` `return` `power(2, n/2 - 1, MOD); ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `n = 6; ` ` ` `cout << countEvenWays(n) << ` `"\n"` `; ` ` ` ` ` `n = 8; ` ` ` `cout << countEvenWays(n); ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Java

`// JAVA program to count ways to write ` `// number as sum of even integers ` ` ` `class` `GFG { ` ` ` ` ` `// Initialize mod variable as constant ` ` ` `static` `int` `MOD = ` `1000000007` `; ` ` ` ` ` `/* Iterative Function to calculate ` ` ` `(x^y)%p in O(log y) */` ` ` `static` `int` `power(` `int` `x, ` `int` `y, ` `int` `p) ` ` ` `{ ` ` ` `// Initialize result ` ` ` `int` `res = ` `1` `; ` ` ` ` ` `// Update x if it is more ` ` ` `// than or equal to p ` ` ` `x = x % p; ` ` ` ` ` `while` `(y > ` `0` `) ` ` ` `{ ` ` ` `// If y is odd, multiply x ` ` ` `// with result ` ` ` `if` `(y % ` `2` `== ` `1` `) ` ` ` `res = (` `1` `* res * x) % p; ` ` ` ` ` `// y must be even now ` ` ` `y = y >> ` `1` `; ` `// y = y/2 ` ` ` `x = (` `1` `* x * x) % p; ` ` ` `} ` ` ` `return` `res; ` ` ` `} ` ` ` ` ` `// Return number of ways to write ` ` ` `// 'n' as sum of even integers ` ` ` `static` `int` `countEvenWays(` `int` `n) ` ` ` `{ ` ` ` `return` `power(` `2` `, n/` `2` `- ` `1` `, MOD); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String args[]) ` ` ` `{ ` ` ` `int` `n = ` `6` `; ` ` ` `System.out.println(countEvenWays(n)); ` ` ` `n = ` `8` `; ` ` ` `System.out.println(countEvenWays(n)); ` ` ` `} ` `} ` ` ` `/* This code is contributed by Nikita Tiwari. */` |

*chevron_right*

*filter_none*

## Python

`# PYTHON program to count ways to write ` `# number as sum of even integers ` ` ` `# Initialize mod variable as constant ` `MOD ` `=` `1e9` `+` `7` ` ` `# Iterative Function to calculate ` `# (x^y)%p in O(log y) ` `def` `power(x, y, p) : ` ` ` `res ` `=` `1` `# Initialize result ` ` ` ` ` `x ` `=` `x ` `%` `p ` `# Update x if it is more ` ` ` `# than or equal to p ` ` ` ` ` `while` `(y > ` `0` `) : ` ` ` ` ` `# If y is odd, multiply x ` ` ` `# with result ` ` ` `if` `(y & ` `1` `) : ` ` ` `res ` `=` `(` `1` `*` `res ` `*` `x) ` `%` `p ` ` ` ` ` `# y must be even now ` ` ` `y ` `=` `y >> ` `1` `# y = y/2 ` ` ` `x ` `=` `(` `1` `*` `x ` `*` `x) ` `%` `p ` ` ` ` ` ` ` `return` `res ` ` ` ` ` `# Return number of ways to write 'n' ` `# as sum of even integers ` `def` `countEvenWays(n) : ` ` ` `return` `power(` `2` `, n` `/` `2` `-` `1` `, MOD) ` ` ` `# Driver code ` `n ` `=` `6` `print` `(` `int` `(countEvenWays(n))) ` `n ` `=` `8` `print` `(` `int` `(countEvenWays(n))) ` ` ` `# This code is contributed by Nikita Tiwari. ` |

*chevron_right*

*filter_none*

## C#

`// C# program to count ways to write ` `// number as sum of even integers ` `using` `System; ` ` ` `class` `GFG { ` ` ` ` ` `// Initialize mod variable as constant ` ` ` `static` `int` `MOD = 1000000007; ` ` ` ` ` `/* Iterative Function to calculate ` ` ` `(x^y)%p in O(log y) */` ` ` `static` `int` `power(` `int` `x, ` `int` `y, ` `int` `p) ` ` ` `{ ` ` ` ` ` `// Initialize result ` ` ` `int` `res = 1; ` ` ` ` ` `// Update x if it is more ` ` ` `// than or equal to p ` ` ` `x = x % p; ` ` ` ` ` `while` `(y > 0) ` ` ` `{ ` ` ` ` ` `// If y is odd, multiply x ` ` ` `// with result ` ` ` `if` `(y % 2 == 1) ` ` ` `res = (1 * res * x) % p; ` ` ` ` ` `// y must be even now ` ` ` `y = y >> 1; ` `// y = y/2 ` ` ` `x = (1 * x * x) % p; ` ` ` `} ` ` ` ` ` `return` `res; ` ` ` `} ` ` ` ` ` `// Return number of ways to write ` ` ` `// 'n' as sum of even integers ` ` ` `static` `int` `countEvenWays(` `int` `n) ` ` ` `{ ` ` ` `return` `power(2, n/2 - 1, MOD); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main() ` ` ` `{ ` ` ` `int` `n = 6; ` ` ` `Console.WriteLine(countEvenWays(n)); ` ` ` ` ` `n = 8; ` ` ` `Console.WriteLine(countEvenWays(n)); ` ` ` `} ` `} ` ` ` `// This code is contributed by vt_m. ` |

*chevron_right*

*filter_none*

## PHP

`<?php ` `// PHP program to count ways ` `// to write number as sum of ` `// even integers ` ` ` `// Initialize mod variable ` `// as constant ` `$MOD` `= 1000000000.0; ` ` ` `/* Iterative Function to ` `calculate (x^y)%p in O(log y) */` `function` `power(` `$x` `, ` `$y` `, ` `$p` `) ` `{ ` ` ` `// Initialize result ` ` ` `$res` `= 1; ` ` ` ` ` `// Update x if it is more ` ` ` `// than or equal to p ` ` ` `$x` `= ` `$x` `% ` `$p` `; ` ` ` ` ` ` ` `while` `(` `$y` `> 0) ` ` ` `{ ` ` ` `// If y is odd, multiply ` ` ` `// x with result ` ` ` `if` `(` `$y` `& 1) ` ` ` `$res` `= (1 * ` `$res` `* ` ` ` `$x` `) % ` `$p` `; ` ` ` ` ` `// y must be even now ` ` ` `$y` `= ` `$y` `>> 1; ` `// y = y/2 ` ` ` `$x` `= (1 * ` `$x` `* ` ` ` `$x` `) % ` `$p` `; ` ` ` `} ` ` ` `return` `$res` `; ` `} ` ` ` `// Return number of ways ` `// to write 'n' as sum of ` `// even integers ` `function` `countEvenWays(` `$n` `) ` `{ ` ` ` `global` `$MOD` `; ` ` ` `return` `power(2, ` `$n` `/ ` ` ` `2 - 1, ` `$MOD` `); ` `} ` ` ` `// Driver code ` `$n` `= 6; ` `echo` `countEvenWays(` `$n` `), ` `"\n"` `; ` ` ` `$n` `= 8; ` `echo` `countEvenWays(` `$n` `); ` ` ` `// This code is contributed ` `// by ajit ` `?> ` |

*chevron_right*

*filter_none*

**Output:**

4 8

**Time complexity: **O(Log(n))

**Auxiliary space: **O(1)

This article is contributed by Shubham Bansal. 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.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the **DSA Self Paced Course** at a student-friendly price and become industry ready.

## Recommended Posts:

- Count ways to express 'n' as sum of odd integers
- Count ways to express a number as sum of consecutive numbers
- Count ways to express a number as sum of exactly two numbers
- Count of different ways to express N as the sum of 1, 3 and 4
- Count of integers in a range which have even number of odd digits and odd number of even digits
- Ways to express a number as product of two different factors
- Number of ways in which N can be represented as the sum of two positive integers
- Number of ways to write N as a sum of K non-negative integers
- Different ways to represent N as sum of K non-zero integers
- Ways to write N as sum of two or more positive integers | Set-2
- Ways to form an array having integers in given range such that total sum is divisible by 2
- Print all possible ways to write N as sum of two or more positive integers
- Number of ways to form a heap with n distinct integers
- Check if a number has an odd count of odd divisors and even count of even divisors
- Median in a stream of integers (running integers)
- Mode in a stream of integers (running integers)
- Lexicographically smallest permutation of size A having B integers exceeding all preceeding integers
- Number of ways to get even sum by choosing three numbers from 1 to N
- Express a number as sum of consecutive numbers
- Express an odd number as sum of prime numbers