# Count ways to express even number ‘n’ as sum of even integers

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; ` `} ` |

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## 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. */` |

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## 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. ` |

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## 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. ` |

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## 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 ` `?> ` |

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**Output:**

4 8

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

**Auxiliary space: **O(1)

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