Consider the relation
$$t_{n}=t+t_{n-1}+\sqrt{(t+t_{n-1})^{2}+t^{2}}, n>1$$
with $t_{1}=(1+\sqrt{2})t$.
Is it possible to get a closed form expression for this relation?
I found that
$$t_{2}=\left(2+\sqrt{2}+\sqrt{7+4\sqrt{2}}\right)t$$
but after this it seems to get harder unless there is a pattern.