x and y are triangular numbers, and we know 8* any triangular number = an odd square.
1+8*(2*y); 1+8*(x+(2*y)); 1+8*(2*(x+y))
1+8*(y); 1+8*(x+y); 1+8*(y+(2*x))
1; 1+8*(x); 1+8*(2*x))

So if 2*x and 2*y are triangular numbers, then you automatically have 5 squared integers.
Then if two out of four possible options of: x+y or 2*(x+y) or x+2*y or y+2*x are triangular numbers. Then you have the 7!