I got this interesting sequence from a friend and I wish to know more about its behavior.
I have a sequence of $10$, $-10$, and $198$ zeroes. Suppose we, for every number in the sequence, replace it with the sum of that number and the number after it in the sequence, last number being replaced with the sum of first and last numbers. We repeat this process indefinitely.
What is the behavior of such a series?
I think that the series will eventually increase indefinitely, looking at smaller case with smaller numbers of zeroes.
original: $10 \ -10 \ \ \ 0 \ \dots \ 0$,
1st time: $0 \ -10 \ \ \ 0 \ \dots \ 10$,
2nd time: $-10 \ -10 \ \ \ 0 \ \dots \ 10 \ \ \ 10$,
3rd time: $-20 \ -10 \ \ \ 0 \ \dots \ 10 \ \ 20 \ \ \ 0$
and so on.