__here__and

__here__) that the intersections, which I call

**H**ouse-

**S**haped

**T**riangular

**N**umbers {OEIS A182427} and

**H**ouse-

**S**haped

**S**quare

**N**umbers {OEIS A214937}, are infinite sets too.

**H**ouse-

**S**haped

**S**quare

**T**riangular

**N**umbers), i.e. how many elements does it consist of? Here I intend to prove the following

**Theorem:**There are infinitely many HSSTNs.

**Proof:**

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

**[(2a+1)^2]*{[(2a+1)^2]+1}*(1/2)**

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

**[(2a+1)^2]*{[(2a+1)^2]+1}*(1/2)=..=**

=8a^4+16a^3+14a^2+6a+1=..=

=8a^4+16a^3+14a^2+6a+1=..=

**=[(2a+1)^2]+8a^4+16a^3+10a^2+2a=..=**

**=[(2a+1)^2]+(4a^2+4a)*(4a^2+4a+1)*(1/2)**

4. On the other hand, this triangular number can be expressed as a square number

**[(2a+1)^2]*{[(2a+1)^2]+1}*(1/2)=..=**

=8a^4+16a^3+14a^2+6a+1=..=

=8a^4+16a^3+14a^2+6a+1=..=

**=((sqrt(a^2+(a+1)^2))*(2a+1))^2**

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)]**