A short Maple program which searches for square-triangular
numbers.  It uses the method some of you suggested:
compute the square root of i*(i+1)/2 and see if it
is an integer.


  N := 1000;
  for i from 1 to N do
     if frac(evalf(sqrt(i*(i+1)/2))) = 0 
        then print(i*(i+1)/2); 
     fi;
  od;


Output of the program:

                              N := 1000

                                  1

                                  36

                                 1225

                                41616