Consider the sequence $s_0=(7,2,1,5,6)$, taking the mean of adjacent terms to form the next sequence $s_1=(\frac{7+2}{2},\frac{2+1}{2},\frac{1+5}{2},\frac{5+6}{2},\frac{6+7}{2})=(4.5,1.5,3,5.5,6.5)$, and similarly $$ s_2=(3,2.25,4.25,6,5.5)\\ s_3=(2.625,3.25,{5.125},\color{red}{5.75},4.25) $$
The terms of the sequence $s_3$ increases first and then decreases. This pattern tends to occur for any sequence of integers after some steps.
ie., $s_i(1)\le s_i(2)\le\cdots \le s_i(k-1)\le s_i(k)\ge s_i(k+1)\cdots$ at least under a cyclic shift, after a finite $i$.
How do I justify such a pattern for a general case of integers ?
Another example: $s_0=(7,1,4,5,0)$ then $$ s_1 = (4, 2.5, 2, 3.5)\\ s_2 = (3.25, 2.25, 2.75, 3.75)\\ s_3 = (2.75, 2.5, 3.25, 3.5)\\ s_4 = (2.625, 2.875, \color{red}{3.375}, 3.125) $$
My Observations
The average of the terms remains the same, in this case $s_i^{avg}=4.2$ for all $i$, which is not difficult to prove. That means, the data are just being redistributed in each step.
$$ s_0=(s_0(0),s_0(2),s_0(3),\cdots,s_0(n-2),s_0(n-1))\\ s_1=(\frac{s_0(0)+s_0(1)}{2},\frac{s_0(1)+s_0(2)}{2},\frac{s_0(2)+s_0(3)}{2},\cdots,\frac{s_0(n-1)+s_0(0)}{2}) $$ Let $d_k(i)=s_k(i+1)-s_{k}(i)$ then $s_{k+1}(i)=\dfrac{s_k(i)+s_k(i+1)}{2}=\dfrac{s_k(i)+s_{k}(i)+d_k(i)}{2}=s_k(i)+\dfrac{d_k(i)}{2}$
and similarly, $d_{k+1}(i)=\dfrac{s_k(i+2)-s_k(i)}{2}$
Generating sequences by taking the mean of the adjacent terms move the terms closer together, ie., over the steps the values must tend to be more uniform around the mean of the sequence.
But, are there anything else causing this pattern to occur?