Let $d_n=a_{n+1}-a_n$ be the difference of two consecutive terms of a sequence of natural numbers.
We can easily construct sequences of natural numbers $a_n$ using trigonometric functions or the floor function which have the property:
*(1)*For infinitely many $n, d_{n+1}<d_n$ .
For example, let $a_n=cos(\frac{n\pi}{2})$ or $a_n=\lfloor \frac{n}{3}\rfloor$ and $n\geq 1$ for both cases.
- My question is
Can anybody give me an example of a sequence having the property (1) but which has a closed formula not containing trigonometric functions,or the floor function?
- Note
I would like to see a closed formula of such a sequence (if possible) and not something general like the primes or a sequence whose sum of reciprosals diverges.
Thank you very much in advance.
EDIT:$(-1)^n=cos(n\pi)$ so don't try something like this. So what i would like to see is something without trigonometric functions ,floor functions or $(-1)^n$.
Everything else is acceptable.