try to solve closed form of recursive equation
$S_{n+1}=\frac{3S_n+(S_n \% 2)}{2}+4$ , $S_1=4$
where "%" is modulo operation, $x\%2$ return $1$ if $x$ is odd else $0$.
This equation is derived from a simple game:
n people with id 0,1,2,...,n-1 sit around a circular table in clockwise,
start a enumeration from man with id 0, they leave the table when enumerated
as multiple of 3 (id 2 always leave first).
find the last man leaving.
for example $a_4=0$, because 2 leave first, and then 1, 3.
solution of it is $a_n=(a_{n-1}+3)\%n$
Series $a_n$ would be like figure 1,
Growing with slope 3, bounded by y=x, drop every time it hit y=x.
Drop points gives a way to calculate $a_n$ with complexity $O(log\;n)$.
And I tried to reduce it to $O(1)$ by finding closed form of drop point $D_i$
$a_3=1, a_4=0$, let 4 be the first drop point, $D_1=4$.
then $S_n=\sum_{i=1}^{n}D_i$ has form I showed.