Let $(T_n)$ be the sequence of Telephone numbers(recall that $ T_0 = T_1 = 1, \ T_{n+1} = T_{n} + nT_{n-1},\ n\ge 1$). Is it true that for any $n\geq 10$, $$ \lfloor \bigg(\frac{T_{n+1}}{T_{n}}\bigg)^2 \rfloor =n+1+\lfloor \sqrt{n}\rfloor \tag{1} $$
(where as usual $\lfloor \ \rfloor$ denotes the "floor" integer part) I have checked this up to $n=25000$. I encountered (1) while working on a recent question