# Pythagorean triplets generated in a unique way

Let's have the following sequence of Pythagorean triplets $$25^2=24^2+7^2,1201^2=1200^2+49^2$$,

$$58825^2=58824^2+343^2, ?, ?$$

What are the next two triplets in this sequence?

How have these triplets been generated?

We have $$7^{2n} = \frac{7^{2n}+1}{2} + \frac{7^{2n}-1}{2} = (\frac{7^{2n}+1}{2} - \frac{7^{2n}-1}{2}) \cdot (\frac{7^{2n}+1}{2} + \frac{7^{2n}-1}{2}) = (\frac{7^{2n}+1}{2})^2 - (\frac{7^{2n}-1}{2})^2$$
$$n=1: 7^2 = 25^2-24^2$$
$$n=2: 7^4 = 49^2 = 1201^2 - 1200^2$$
$$n=3: 7^6 = 343^2 = 58825^2 - 58824^2$$
$$n=4: 7^8 = 2401^2 = 2882401^2 - 2882400^2$$
$$n=5: 7^{10} = 16807^2 = 141237625^2 - 141237624^2$$