Example of a sequence such that $n(a_n-a_{n-2}) + b n(a_n-a_{n-1}) \to 0$ but $n(a_n-a_{n-2}) \nrightarrow 0$.

Question

Let $b \neq 0$. Do you know some example of a sequence such that $n(a_n-a_{n-2}) + b n(a_n-a_{n-1}) \to 0$ but $n(a_n-a_{n-2}) \nrightarrow 0$?

Edit: I also need that $a_n \to 0$.

Answer 1

Let $b=-{1\over 2},$ $a_n=(-2)^n.$ Then $$n(a_n-a_{n-2})+bn(a_n-a_{n-1})=0$$ and $$n(a_n-a_{n-2})=(-1)^nn[2^n-2^{n-2}]$$