Special Pythagorean Triples

$(430,1824,1874)$ is a special Pythagorean triple.
Indeed, we have:
${430}^2+2133331200={46190}^2$
${1824}^2+2133331200={46224}^2$
${1874}^2+2133331200={46226}^2$

We call special a Pythagorean triple if there exists a positive integer $t$ such as for each element $e$ of the triple $e^2+t$ is a square.

There can exist multiple integers for a given triple.
For instance, the triple $(24531,30600,39219)$ is special with the values $133453264$ and $119200745664$.

Find all special Pythagorean triples $(a,b,c)$ with $a+b+c<2000000$
(a triple is counted for each distinct $t$)

Answer format: $Count,\sum t$

You are given: $18,17172343737$ for a threshold of $10000$

[My timing: < 1 min. ]