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

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

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 implemnetation 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 
?> 

Output:

Yes

Time Complexity: O(Sqrt(N))

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My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python 3

C#

PHP

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