> There may be a recurrence that does all three at once, not sure.

Now that we know the start of the sequence, we can just dump it into OEIS to look up the answer! :)

The sequence is A046180 (https://oeis.org/A046180) titled "Hexagonal pentagonal numbers" with a nice and easy recurrence relation:

    a(n) = 37635*a(n-1) - 37635*a(n-2) + a(n-3).

Also, according to the comments on OEIS, all hexagonal numbers are triangular, so we could have just skipped that requirement entirely.