There is a random walk on a linear lattice of size $\{0,N\}$, where $0$ is the origin and a reflecting boundary and $N$ is the absorbing boundary. It moves forward or backward one step at a time with $\frac12$ probability. The number of times a state/point $(p)$ inside the lattice being visited before it gets absorbed was calculated as $2(N-p)$, $2$ times the distance between the point and $N$. (I got this result in simulation, I don't know how to derive this mathematically.)
If I increase the step size (moving more than one step at a time) the walker can take $x+i$ or $x-i$ with $\frac{1}{2}$ probability, where $i=1,2,3,\dots,k$. That is if k is 2, then the walker can either take 1 step or 2 step forward/backward with equal probability ($\frac{1}{2k}$).
However, if the walker goes beyond the boundary [$<0$ or $>N$], then it'll be reflected back as follows, $$ \mbox{new x} = \begin{Bmatrix} 0+(0-x) & \mbox{if} \ x < 0\\ N-(x-N) & \mbox{if} \ x > N \end{Bmatrix} $$
How to derive the number of visits [i.e. $2(N-p)$] mathematically for single step random walk? And how the increase in step size $k$ will affect the same?
Thanks in advance.