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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.

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