# Second question on a real sequence

I am thinking again about the real sequence $$\{a_n\}_{n\ge1}$$ which decays faster than any algebraic speed, that is, $$\lim_{n\to \infty}n^pa_n = 0$$ for every integer $$p$$. Actually, $$a_n$$ can be treated as the Fourier sequence of a smooth periodic function. Assume now the following two equations hold for every integer $$k\ge 0$$:

1. $$\sum_{n\ge 1}n^k(n+1)^k(\frac{n+1}{n}-\frac{n}{n+1})a_na_{n+1} = 0$$;
2. $$\sum_{n\ge 1}n^k(n+2)^k(\frac{n+2}{n}-\frac{n}{n+2})a_na_{n+2} = 0$$.

In this case, can we conclude that $$a_na_{n+1} = 0$$ and $$a_na_{n+2} = 0$$? I already know from the discussion in A question on a real sequence that only one of the above assumption does not lead to the conclusion. I am not sure what will happen if we combine these two conditions together.