To find this sequence, if it exists, you would need to find a set of Heronian triangles such that the lengths of sides a:b:c in each triangle corresponded to sides b:c:d in the next, until the last triangle in the (hypothetical 26 triangle) set had sides z:(an):(bn), which would then be followed by (an):(bn):(c*n). A cursory search of the hundred smallest integer Heronian triangles yields no such cycle.

To find this sequence, if it exists, you would need to find a set of Heronian triangles such that the lengths of sides a:b:c in each triangle corresponded to sides b:c:d in the next. The series of triangle proportions could (and likely will, if it exsts, I think) contain a cycle wherein a multiplier is introduced after each cycle, such that the triangles after x:y:z are y:z:(an), z:(an):(bn), and (an):(bn):(cn).

A cursory search of the hundred smallest integer Heronian triangles yields no such set longer than 2 triangles, and no such cycle. As Heronian triangles can be parametrically enumerated, it would be possible to perform a brute force search of a sizable number of them for such a sequence or cycle.