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Check if a number can be represented as a sum of 2 triangular numbers

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  • Difficulty Level : Medium
  • Last Updated : 23 Jun, 2022

Given an integer N, the task is to find out whether it can be written as a sum of 2 triangular numbers (which may or may not be distinct).
Examples: 
 

Input: N = 24
Output: YES
24 can be represented as 3+21.

Input: N = 15
Output: NO

 

Approach: 
Consider all triangular numbers less than N i.e. sqrt(N) numbers. Let’s add them to a set, and for each triangular number X we check if N-X is present in the set. If it is true with any triangular number, then the answer is YES, otherwise the answer is NO.
Below is the implementation of the above approach: 
 

C++




// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to check if it is possible or not
bool checkTriangularSumRepresentation(int n)
{
    unordered_set<int> tri;
    int i = 1;
 
    // Store all triangular numbers up to N in a Set
    while (1) {
        int x = i * (i + 1) / 2;
        if (x >= n)
            break;
        tri.insert(x);
        i++;
    }
  
    // Check if a pair exists
    for (auto tm : tri)
        if (tri.find(n - tm) != tri.end())
            return true;
    return false;
}
 
// Driver Code
int main()
{
    int n = 24;
    checkTriangularSumRepresentation(n) ? cout << "Yes"
                                        : cout << "No";
 
    return 0;
}

Java




// Java implementation of the approach
import java.util.*;
 
class GFG
{
 
    // Function to check if it is possible or not
    static boolean checkTriangularSumRepresentation(int n)
    {
        HashSet<Integer> tri = new HashSet<>();
        int i = 1;
 
        // Store all triangular numbers up to N in a Set
        while (true)
        {
            int x = i * (i + 1) / 2;
            if (x >= n)
            {
                break;
            }
            tri.add(x);
            i++;
        }
 
        // Check if a pair exists
        for (Integer tm : tri)
        {
            if (tri.contains(n - tm) && (n - tm) !=
                (int) tri.toArray()[tri.size() - 1])
            {
                return true;
            }
        }
        return false;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int n = 24;
        if (checkTriangularSumRepresentation(n))
        {
            System.out.println("Yes");
        }
        else
        {
            System.out.println("No");
        }
    }
}
 
// This code has been contributed by 29AjayKumar

Python 3




# Python3 implementation of the above approach
 
# Function to check if it is possible or not
def checkTriangularSumRepresentation(n) :
     
    tri = list();
    i = 1;
 
    # Store all triangular numbers
    # up to N in a Set
    while (1) :
        x = i * (i + 1) // 2;
        if (x >= n) :
            break;
             
        tri.append(x);
        i += 1;
 
    # Check if a pair exists
    for tm in tri :
        if n - tm in tri:
            return True;
             
    return False;
 
# Driver Code
if __name__ == "__main__" :
    n = 24;
     
    if checkTriangularSumRepresentation(n) :
        print("Yes")
    else :
        print("No")
 
# This code is contributed by Ryuga       

C#




// C# implementation of the approach
using System;
using System.Collections.Generic;
 
class GFG
{
 
    // Function to check if it is possible or not
    static bool checkTriangularSumRepresentation(int n)
    {
        HashSet<int> tri = new HashSet<int>();
        int i = 1;
 
        // Store all triangular numbers up to N in a Set
        while (true)
        {
            int x = i * (i + 1) / 2;
            if (x >= n)
            {
                break;
            }
            tri.Add(x);
            i++;
        }
 
        // Check if a pair exists
        foreach (int tm in tri)
        {
            if (tri.Contains(n - tm))
            {
                return true;
            }
        }
        return false;
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        int n = 24;
        if (checkTriangularSumRepresentation(n))
        {
            Console.WriteLine("Yes");
        }
        else
        {
            Console.WriteLine("No");
        }
    }
}
 
// This code contributed by Rajput-Ji

PHP




<?php
// PHP implementation of the above approach
 
// Function to check if it is possible or not
function checkTriangularSumRepresentation($n)
{
    $tri = array();
    $i = 1;
 
    // Store all triangular numbers
    // up to N in a Set
    while (true)
    {
        $x = $i * ($i + 1);
        if ($x >= $n)
            break;
             
        array_push($tri, $x);
        $i += 1;
    }
     
    // Check if a pair exists
    foreach($tri as $tm)
        if (in_array($n - $tm, $tri))
            return true;
             
    return false;
}
 
// Driver Code
$n = 24;
 
if (checkTriangularSumRepresentation($n))
    print("Yes");
else
    print("No");
 
// This code is contributed by mits
?>

Javascript




<script>
 
// Javascript implementation of the above approach
 
// Function to check if it is possible or not
function checkTriangularSumRepresentation(n)
{
    var tri = new Set();
    var i = 1;
 
    // Store all triangular numbers up to N in a Set
    while (1) {
        var x = i * parseInt((i + 1) / 2);
        if (x >= n)
            break;
        tri.add(x);
        i++;
    }
     
    var ans = false;
    // Check if a pair exists
    tri.forEach(tm => {
        if (tri.has(n - tm))
            ans = true
    });
    return ans;
}
 
// Driver Code
var n = 24;
checkTriangularSumRepresentation(n) ? document.write( "Yes")
                                    : document.write( "No");
 
// This code is contributed by famously.
</script>

Output: 

Yes

 

Time Complexity: O(Sqrt(N))

Auxiliary Space: O(sqrt(N))

Second approach : Its very clear that N is positive because triangular numbers are numbers that are representable as k*(k+1)/2  where k is some positive integer by this we also get that each term will be less than N. This means that we take and iterate over one of the other terms.  If we iterate over the first term in increasing order, then we can use two pointers, which gives us a solution in O(Sqrt(N)) .

C++




// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
#define ll long long
int main()
{
    ll n;
    cin >> n;
    ll dig = 2 * n;
    bool flag= 0;
    for (ll i = 1; i < sqrt(2 * n + 1); i++) {
        if (i * i + i > dig)
            break;
        ll rest = dig - i * i - i;
        ll left = 1;
        ll right = dig;
        while (left <= right) {
            ll mid = (left + right) / 2;
            if (mid * mid + mid > rest)
                right = mid - 1;
            else if (mid * mid + mid == rest) {
                flag = 1;
                break;
            }
            else
                left = mid + 1;
        }
        if (flag)
            break;
    }
    if (flag)
        cout<<"YES"<<endl;
    else
        cout<<"NO"<<endl;
 
    return 0;
}
 
 
// this code is contributed by vaibhavsinghii

Java




// Java implementation of the above approach
import java.util.*;
class GFG {
 
  public static void main(String[] args) {
    int n = 24;
 
    int dig = 2 * n;
    boolean flag = false;
    for (int i = 1; i < Math.sqrt(2 * n + 1); i++) {
      if (i * i + i > dig)
        break;
      int rest = dig - i * i - i;
      int left = 1;
      int right = dig;
      while (left <= right) {
        int mid = (left + right) / 2;
        if (mid * mid + mid > rest)
          right = mid - 1;
        else if (mid * mid + mid == rest) {
          flag = true;
          break;
        } else
          left = mid + 1;
      }
      if (flag)
        break;
    }
    if (flag)
      System.out.print("YES");
    else
      System.out.print("NO");
 
  }
}
 
// This code is contributed by Rajput-Ji

Python3




# Python implementation of the above approach
import math
 
if __name__ == '__main__':
    n = 24;
 
    dig = 2 * n;
    flag = False;
    for i in range(1, int(math.sqrt(2 * n + 1))):
        if (i * i + i > dig):
            break;
        rest = dig - i * i - i;
        left = 1;
        right = dig;
        while (left <= right):
            mid = (left + right) // 2;
            if (mid * mid + mid > rest):
                right = mid - 1;
            elif(mid * mid + mid == rest):
                flag = True;
                break;
            else:
                left = mid + 1;
         
        if (flag):
            break;
     
    if (flag):
        print("YES");
    else:
        print("NO");
 
# This code is contributed by Rajput-Ji

C#




// C# implementation of the above approach
using System;
public class GFG
{
 
  public static void Main(String[] args)
  {
    int n = 24;
 
    int dig = 2 * n;
    bool flag = false;
    for (int i = 1; i < Math.Sqrt(2 * n + 1); i++)
    {
      if (i * i + i > dig)
        break;
      int rest = dig - i * i - i;
      int left = 1;
      int right = dig;
      while (left <= right) {
        int mid = (left + right) / 2;
        if (mid * mid + mid > rest)
          right = mid - 1;
        else if (mid * mid + mid == rest) {
          flag = true;
          break;
        } else
          left = mid + 1;
      }
      if (flag)
        break;
    }
    if (flag)
      Console.Write("YES");
    else
      Console.Write("NO");
 
  }
}
 
// This code is contributed by Rajput-Ji

Javascript




<script>
// javascript implementation of the above approach
        var n = 24;
 
        var dig = 2 * n;
        var flag = false;
        for (i = 1; i < Math.sqrt(2 * n + 1); i++) {
            if (i * i + i > dig)
                break;
            var rest = dig - i * i - i;
            var left = 1;
            var right = dig;
            while (left <= right) {
                var mid = parseInt((left + right) / 2);
                if (mid * mid + mid > rest)
                    right = mid - 1;
                else if (mid * mid + mid == rest) {
                    flag = true;
                    break;
                } else
                    left = mid + 1;
            }
            if (flag)
                break;
        }
        if (flag)
            document.write("YES");
        else
            document.write("NO");
 
// This code is contributed by Rajput-Ji
</script>

Output:

YES

Time Complexity: O(Sqrt(N))


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