Beautiful problem

2018 Netherlands TST 2019 Peru TST

I'll post a quick sketch to proving the finitenesss (the actual description is just a little bit of casework after the conclusion of the proof for finiteness) The idea is to write $b_i^2 = a_i+a_{i-1}$ and proving that $b_{i+1} = b_i + 1$ by using simple bounding. Once you have done this, the number $k = \lfloor \sqrt{2a_0} \rfloor$ will become important and you will get that all differences of the form $2\ell+(k+1)^2-a_0$ and $2\ell+a_0-k^2$ are achievable. This essentially tells you that all large enough odd numbers and all large enough even numbers are expressible; thereby solving the problem. (for the actual construction one needs some more rigour but it's not that hard to recreate; one can prove these are the smallest possible differences and nothing below these can be constructed)