There are $N$ children sitting along a circle, numbered $1,2,\dots,n$ clockwise. The ith child has a piece of paper with number $a_i$ written on it. They play the following game:
In the first round, the child numbered $x$ adds to his number the sum of the numbers of his neighbors. In the second round, the child next in clockwise order adds to his number the sum of the numbers of his neighbors, and so on. The game ends after M rounds have been played.
any idea about how to get the value of $j$-th element when the game is started at the $i$-th position after $M$ rounds.
is there any closed form equation???