Theorem: There are infinitely many HSSTNs.
1. There exist infinitely many Pythagorean triples (a, b , c), where b=a+1. The sum of a and b can be expressed as 2a+1. Therefore, there are infinitely many such sums (OEIS A002315).
2. Let us take the n-th such sum, i.e. OEIS A002315(n), find the numbers a and b, square their sum 2a+1, and build a triangular number. The result will look like
3. On the one hand, we find that this triangular number can be expressed as a sum of a square number and a triangular number
4. On the other hand, this triangular number can be expressed as a square number
5. Therefore, for any a>1, i.e. for any n>0, there are infinitely many numbers that are simultaneously square, triangular and expressible as a sum a positive square number and a positive number. QED
In fact, any number of the form OEIS A001110(2n+1), where n>0, is such a number. Are there more HSSTNs? Yes, there are. For example, A001110(6) is such a number as A001110(6)=48,024,900=(6,918^2)+[576*577*(1/2)]