How many squares of all sizes arise using an $n$-by-$n$ checkerboard? How many triangles of all sizes arise using a triangular grid with sides of length $n$ ?
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It is very simple to see how many 1-squares fit. What about a 2-square (natural notation)? Well, let's only place the top-left square. How many places can we put the top-left square of a 2 by 2 on the board and have it fit? And so on? This leads to your intuition being correct.
This works for triangles too - but I'll let you figure that out.
For the triangular grid, can you convince yourself that all the triangles are equilateral (assuming the grid is)?
Your answer for the squares is correct. For the triangles, you do something similar, with a twist.
For the triangles in the original direction you have 1 big triangle, 3 triangles one size smaller, 6 another size smaller and you should be able to persuade yourself that these continue as the triangle numbers $1,3,6,10,15,21,\ldots$.
For the triangles in the other direction, if you have an original triangle with an even length side you have 1 triangle with side half that of the original triangle, 6 triangles one size smaller, 15 another size smaller etc., while if you have an original triangle with an odd length side you have 3 triangles with side half that of the original triangle rounded down, 10 triangles one size smaller, 21 another size smaller etc.
Adding all these up is not trivial, but you should get OEIS A002717 as your result.