Nested functional equation: $ f \big( f ( n ) \big) = n ^ 2 $

Question

Does there exist a function $ f : \mathbb N \to \mathbb N $ such that $ f \big( f ( n ) \big) = n ^ 2 $?

Is there a way to do this by considering the fixed points, $ 0 $ and $ 1 $?

Answer 1

Take $f(0)=0$ and $f(1) = 1$. Enumerate all members of $\mathbb N$ that are not squares as $m_j$, $j \ge 1$, then take $$ \eqalign{f(m_{j}^{ 2^k}) &= m_{j+1}^{ 2^k} \ \text{if $j$ is odd}\cr f(m_{j}^{ 2^k}) &= m_{j-1}^{ 2^{k+1}}\ \text{if $j$ is even}\cr} $$