Number of integer points inside an equilateral triangle with side length $n\in\mathbb{N}$

Question

Does there exist any $n\in\mathbb{N}$ such that there exist at least one point inside of(not on the border) an equilateral triangle with side length $n$ ,which its distances to the vertices be integers?

If yes; can anyone give a formula for the number of such points for each $n\in\mathbb{N}$ ?

Notice: We already know by this post and for further results here ,that there are infinite number of points with rational distances inside an equilateral triangle with side length one, but what about integers?

Answer 1

Yes, infinitely many such triangles exist. Possible n are 112, 147, 185, 224, 273, 283, 294, 331...... See https://oeis.org/A061281