Count ways to express ‘n’ as sum of odd integers

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

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 nth fibonnaci 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#

// 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.
?>

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.

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.






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