Can I bound or generate within some distance, the sequence 24, 120, 240, 336, 840, ...?

Question

I am currently examining the sequence A198438. It is defined as:

"Common differences in triples of squares in arithmetic progression, that are not a multiples of other triples in (A198384, A198385, A198386)."

Here are some terms,

24, 120, 240, 336, 840, 840, 720, 1320, 2520, 2016, 3696, 2184, 5280, 5544, 6240, 3960, 9360, 10920, 10296, 4896, 6864, 14280, 18480, 6840, 22440, 17160, 10920, 20064

I'm fairly certain there is no way to generate this algebraically (if I'm mistaken that would be awesome and please let me know) but what I'd like to know is some way to bound it algebraically. For example I'd like to say something like this,

Let $a_n = A198438$,

$a_n < 40n^2$ (This may actually be true)

Or better yet bounded between something like,

$n^2 < a_n < 40n^2$

To summarize, is there a way to bound this sequence in a closed form like above or generate it algebraically? If so, what is the least upper bound you can find and the greatest lower bound?

Edit: To make it a little more general I'm interested in what we can say about the $n$th term in the sequence. Also when I say algebraic I mean it a little more loosely for example, is there an algorithm that can compute this sequence exactly?

Edit2: Will Jagy gave some phenomenal insight to this but I'd still like to know: How do I go from my triples to the sequence and is there a way to chose m and n so it comes out in that order?

Answer 1

Pythagorean triples (primitive) are $$ m^2 - n^2, 2mn, m^2 + n^2. $$

Your triples are $$ m^2 + 2mn - n^2 \; , \; \; m^2 - 2mn - n^2 \; , \; \; m^2 + n^2 $$

For the common difference of the squares, I get, with $m > n$ and $m+n$ odd, and $\gcd(m,n) = 1,$ $$ 4mn \left( m^2 - n^2 \right) $$

Putting these in order is not immediate:

Thu Oct 31 20:31:04 PDT 2019
        24       2    1
       120       3    2
       240       4    1
       336       4    3
       720       5    4
       840       5    2
       840       6    1
      1320       6    5
      2016       8    1
      2184       7    6
      2520       7    2
      3360       8    7
      3696       7    4
      3960      10    1
      4896       9    8
      5280       8    3
      5544       9    2
      6240       8    5
      6840      10    9
      6864      12    1
      9240      11   10
      9360       9    4
     10296      11    2
     10920      10    3
     10920      14    1
     12144      12   11
     14280      10    7
     15600      13   12
     17160      13    2
     18480      11    4
     19656      14   13
     20064      11    8
     22440      11    6
     24360      15   14
     26520      15    2
     28560      12    5
     31416      14    3
     31824      13    4
     31920      12    7
     35880      13   10
     41496      13    6
     43680      13    8
     46200      14   11
     47880      14    5
     50160      15    4
     57960      14    9
     77280      15    8
jagy@phobeusjunior:~$