I believe that the answer to the question is negative, because if it were possible, we would be generally able to tell if some positive integer is prime or not using the sieve of Sundaram, but I would like to know additional arguments (or counterarguments!) supporting a negative answer.
To be more especific, with "generally" I mean using some general method to show that some given positive integer is not of the form mentioned. The question could be rephrased as how to find the positive integers which are not solutions to the equation $x+y+2xy=z$, being $x,y,z$ positive integers.
Thanks in advance!