I'm interested in finite sequences of positive integers $(a_1, a_2, \cdots, a_n)$ such that $$ \sqrt{(a_1, a_2, \cdots, a_n)} := \sqrt{a_1+\sqrt{a_2 + \cdots + \stackrel{\vdots}{\sqrt{a_n}}}} $$ is an integer. Call such a sequence "square-rootable".
If is a sequence is square-rootable, does that imply all of its suffixes are too? That is, must $\sqrt{(a_i, a_{i+1}, \dots a_n)}$ also be an integer for all $1 \leq i \leq n$? If not, what is a counter-example?
If $\sqrt{(a_1,\ldots,a_n)}\in\mathbb{N}$ then $$\sqrt{(a_i,\ldots,a_n)}=(\ldots((\sqrt{(a_1,\ldots,a_n)}^2-a_1)^2-a_2)^2-\ldots)^2-a_{i-1}\in\mathbb{N}$$