Count ways to express 'n' as sum of odd integers

Last Updated : 12 Oct, 2022

Given an positive integer n. Count total number of ways to express 'n' as sum of odd positive integers.

Examples:

Input: 4
Output: 3
Explanation
There are only three ways to write 4
as sum of odd integers:
1. 1 + 3
2. 3 + 1
3. 1 + 1 + 1 + 1

Input: 5
Output: 5

Simple approach is to find recursive nature of problem. The number 'n' can be written as sum of odd integers from either (n-1)^th number or (n-2)^th number. Let the total number of ways to write 'n' be ways(n). The value of 'ways(n)' can be written by recursive formula as follows:

ways(n) = ways(n-1) + ways(n-2)

The above expression is actually the expression for Fibonacci numbers. Therefore problem is reduced to find the n^th fibonacci number.

ways(1) = fib(1) = 1
ways(2) = fib(2) = 1
ways(3) = fib(2) = 2
ways(4) = fib(4) = 3

C++

// C++ program to count ways to write
// number as sum of odd integers
#include<iostream>
using namespace std;

// Function to calculate n'th Fibonacci number
int fib(int n)
{
  /* Declare an array to store Fibonacci numbers. */
  int f[n+1];
  int i;

  /* 0th and 1st number of the series are 0 and 1*/
  f[0] = 0;
  f[1] = 1;

  for (i = 2; i <= n; i++)
  {
      /* Add the previous 2 numbers in the series
         and store it */
      f[i] = f[i-1] + f[i-2];
  }

  return f[n];
}

// Return number of ways to write 'n'
// as sum of odd integers
int countOddWays(int n)
{
    return fib(n);
}

// Driver code
int main()
{
    int n = 4;
    cout << countOddWays(n) << "\n";

    n = 5;
    cout << countOddWays(n);
   return 0;
}

Java

// Java program to count ways to write
// number as sum of odd integers
import java.util.*;

class GFG {
    
// Function to calculate n'th Fibonacci number
static int fib(int n) {
    
    /* Declare an array to store Fibonacci numbers. */
    int f[] = new int[n + 1];
    int i;

    /* 0th and 1st number of the series are 0 and 1*/
    f[0] = 0;
    f[1] = 1;

    for (i = 2; i <= n; i++) {
        
    /* Add the previous 2 numbers in the series
        and store it */
    f[i] = f[i - 1] + f[i - 2];
    }

    return f[n];
}

// Return number of ways to write 'n'
// as sum of odd integers
static int countOddWays(int n)
{
    return fib(n);
}

// Driver code
public static void main(String[] args) {
    
    int n = 4;
    System.out.print(countOddWays(n) + "\n");

    n = 5;
    System.out.print(countOddWays(n));
}
}

// This code is contributed by Anant Agarwal.

Python3

# Python code to count ways to write
# number as sum of odd integers

# Function to calculate n'th 
# Fibonacci number
def fib( n ):

    # Declare a list to store 
    # Fibonacci numbers.
    f=list()
    
    # 0th and 1st number of the 
    # series are 0 and 1
    f.append(0)
    f.append(1)
    
    i = 2
    while i<n+1:

        # Add the previous 2 numbers 
        # in the series and store it
        f.append(f[i-1] + f[i-2])
        i += 1
    return f[n]

# Return number of ways to write 'n'
# as sum of odd integers
def countOddWays( n ):
    return fib(n)

# Driver code
n = 4
print(countOddWays(n))
n = 5
print(countOddWays(n))

# This code is contributed by "Sharad_Bhardwaj"

// C# program to count ways to write
// number as sum of odd integers
using System;

class GFG {
    
    // Function to calculate n'th
    // Fibonacci number
    static int fib(int n) {
        
        /* Declare an array to store
        Fibonacci numbers. */
        int []f = new int[n + 1];
        int i;
    
        /* 0th and 1st number of the
        series are 0 and 1*/
        f[0] = 0;
        f[1] = 1;
    
        for (i = 2; i <= n; i++)
        {
            
            /* Add the previous 2 numbers
            in the series and store it */
            f[i] = f[i - 1] + f[i - 2];
        }
    
        return f[n];
    }
    
    // Return number of ways to write 'n'
    // as sum of odd integers
    static int countOddWays(int n)
    {
        return fib(n);
    }
    
    // Driver code
    public static void Main()
    {
        int n = 4;
        Console.WriteLine(countOddWays(n));
    
        n = 5;
        Console.WriteLine(countOddWays(n));
    }
}

// This code is contributed by vt_m.

PHP

<?php
// PHP program to count ways to write
// number as sum of odd integers

// Function to calculate n'th
// Fibonacci number
function fib($n)
{
    
    // Declare an array to 
    // store Fibonacci numbers.
    $f = array();
    $i;
    
    // 0th and 1st number of the
    // series are 0 and 1
    $f[0] = 0;
    $f[1] = 1;
    
    for($i = 2; $i <= $n; $i++)
    {
        
        // Add the previous 2 
        // numbers in the series
        // and store it 
        $f[$i] = $f[$i - 1] + 
                 $f[$i - 2];
    }
    
    return $f[$n];
}

// Return number of ways to write 'n'
// as sum of odd integers
function countOddWays( $n)
{
    return fib($n);
}

    // Driver Code
    $n = 4;
    echo countOddWays($n) , "\n";
    $n = 5;
    echo countOddWays($n);
    
// This code is contributed by anuj_67.
?>

JavaScript

<script>

// Javascript program to count ways to write
// number as sum of odd integers

// Function to calculate n'th Fibonacci number
function fib(n) {
      
    /* Declare an array to store Fibonacci numbers. */
    let f = [];
    let i;
  
    /* 0th and 1st number of the series are 0 and 1*/
    f[0] = 0;
    f[1] = 1;
  
    for (i = 2; i <= n; i++) {
          
    /* Add the previous 2 numbers in the series
        and store it */
    f[i] = f[i - 1] + f[i - 2];
    }
  
    return f[n];
}
  
// Return number of ways to write 'n'
// as sum of odd integers
function countOddWays(n)
{
    return fib(n);
}
    
// Driver code
        let n = 4;
    document.write(countOddWays(n) + "<br/>");
  
    n = 5;
    document.write(countOddWays(n));
    
    // This code is contributed by code_hunt.
</script>

Output:

3
5

Note: The time complexity of the above implementation is O(n). It can be further optimized up-to O(Logn) time using Fibonacci function optimization by Matrix Exponential.
Auxiliary Space: O(n)

Count ways to express a number as sum of exactly two numbers

Shubham Bansal

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Count ways to express 'n' as sum of odd integers

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