The sequence is known to be infinite.
Once sufficiently many further terms are found, their indices in A274306 can give another sequence.
The terms' indices in A274306 are: 0, 3, 20, 119, 696, 4059, 23660, 137903, ... - Chai Wah Wu, Jun 20 2016
The terms' indices are equal to the sequence A001652 and thus satisfy a linear recurrence. Gürel showed that 2k^2+2k+1 = k^2 + (k+1)^2 is a square if and only if A274306(k) is a square, i.e, (k, k+1) are the 2 smaller terms of a Pythagorean triple and thus k is the index of a term if and only if it is in A001652. - Chai Wah Wu, Jun 21 2016