The sequence $(U_n)_{n≥0}$ is defined by $U_0 = 1$ and $∀n ∈ \mathbb{N}$* : $U_n = U_{⌊n/2⌋} + U_{⌊n/3⌋} + U_{⌊n/6⌋}$.
a) Show that $∀n ∈ \mathbb{N}$, $U_n ≥n+1$.
b) Find the minimal value for $C > 0$ such that $∀n ∈ \mathbb{N}$: $U_n ≤C(n+1)$.
for question a) I tried some strong induction to say that $⌊n/2⌋+⌊n/3⌋+⌊n/6⌋≥n-2$ and prove that $⌊n+1/2⌋+⌊n+1/3⌋+⌊n+1/6⌋≥n-1$. I replace $n$ by $6k,6k+1,...,6k+5$ with $k ∈ \mathbb{N}$ and it seems to be correct don't know if it is ? But I don't know how to do the question b) because if I applied the same reasoning. I would get something like $⌊n+1/2⌋+⌊n+1/3⌋+⌊n+1/6⌋\leq n-1$ I'm confusing I know I made it wrong and maybe also in the first part