Question 64
A sequence $(u_n)_{n\ge1}$ is called Sicilian if $u_1$ is a positive integer and, for every $n$,
$$u_{n+1}=\begin{cases}u_n/2&\text{if }u_n\text{ is even},\\u_n+\lfloor\sqrt{u_n}\rfloor&\text{if }u_n\text{ is odd}.\end{cases}$$
Does there exist a Sicilian sequence with $u_n\gt 1$ for every $n$? (Here $[x]=\lfloor x\rfloor$ denotes the integer part; for example $[2.71828]=2$.)
$$u_{n+1}=\begin{cases}u_n/2&\text{if }u_n\text{ is even},\\u_n+\lfloor\sqrt{u_n}\rfloor&\text{if }u_n\text{ is odd}.\end{cases}$$
Does there exist a Sicilian sequence with $u_n\gt 1$ for every $n$? (Here $[x]=\lfloor x\rfloor$ denotes the integer part; for example $[2.71828]=2$.)