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Edit: so it turns out these are called square triangular numbers

I was helping my sister out with her APCS homework and one of the questions was to find the first $n$ "magic squares" where a magic square is defined as a number that can be expressed as $\sum_{i=0}^x i = \frac {x (x+1)} 2$ and $y^2$ for some integers $x, y$, with the first 3 magic squares being 1, 36, and 1225. Yes, I know this is not the typical definition of magic square, and this was probably done to make the question less googlable.

Just for fun, I tried to find the most optimal solution. My final solution, although I have absolutely no idea how it works, is this:

n = 69

a = 1
b = 1
magic = [1]
while len(magic) < n:
    b = 2 * a + b
    a = b - a
    magic.append(a * a * b * b)

I reached this solution because I was initally going through all the squares and checking if double the square minus 1 or double the square plus 1 was also a square (which would satisfy the equation $n^2 + n - 2 a^2 = 0$). For example, looking at the square 4, we see that $4 \times 2 + 1 = 9$ is a square, so the equation is satisfied for $a=36$. Looking at the squares that ended up producing magic square, the magic square ended up being $1 \times 1 = 1^2 \times 1^2$, $4 \times 9 = 2^2 \times 3^2$, $25 \times 49 = 5^2 \times 7^2$, $144 \times 289 = 12^2 \times 19^2$, etc, which follows the pattern described in my code. Why?

And yeah I've never used math stackexchange before so let me know if I should be asking this differently.

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  • $\begingroup$ Look it up at OEIS, they usually have all the information oeis.org/A001110 $\endgroup$ – Yuriy S Oct 23 '18 at 21:37
  • $\begingroup$ wow, that's super helpful, i'll see if this is in there somewhere $\endgroup$ – Aayush Tyagi Oct 23 '18 at 22:07

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