Taxicab Numbers Extended: Extended

I liked my extension of taxicab numbers so much that I've composed an addition to it.

These three formulas are ones I came up with to generate an infinite number of the numbers I talked about last time.

5(2n^2+2n+1)
5(n^2+4n+5)
13(2n^2+10n+13)

The first only works for n greater than or equal two, whereas the other two work for positive integral n.

These formulas were found by looking at the problem as a difference of squares problem, and from there as this: Finding two strings of consecutive odd integers that are equal.

The third formula also allowed me to disprove one of my earlier conjectures; that all of the numbers are multiples of 5. This formula found a whole set of them, the smallest of which is 533. (23^2 + 2^2 and 22^2 + 7^2.)

There is still more to do, since I haven't yet found a formula that will determine every single number of this type, though having an infinite number of them is, well, pretty damn cool.