jasperE3 wrote: Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and $$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$. Constant $P(x)$ implies constant $f(x)$ which is never a solution So let us consider that highest summand of $P(x)$ is $ax^n$ with $a\ne 0$ and $n\ge 1$ For $x$ great enough, we obviously have $|f(f(x))|\sim |a|^{n+1}x^{n^2}$ and so $n=1$ (else $|f(f(x))>>|4f(x)-x|$ And so $P(x)=ax$ and we have $\lfloor a\lfloor ax\rfloor\rfloor+x=4\lfloor ax\rfloor$ Taking once again $x$ great enough, we need $a^2+1=4a$ and so $a=2-\sqrt 3$ or $a=2+\sqrt 3$ Easy to check that the first does not fit and less easy (but possible) to check that second fits. Hence the answer $\boxed{P(x)=(2+\sqrt 3)x}$