  Parallel Diffusion is a variant of Chip-Firing introduced in 2018 by Duffy et
al. In Parallel Diffusion, chips move from places of high concentration to
places of low concentration through a discrete-time process. At each time step,
every vertex sends a chip to each of its poorer neighbours, allowing for some
vertices to perhaps fall into debt (represented by negative stack sizes). In
their recent paper, Long and Narayanan proved a conjecture from the original
paper by Duffy et al. that every Parallel Diffusion process eventually, after
some pre-period, exhibits periodic behaviour. With this result, we are now able
to count the number of these periods that exist up to a definition of
isomorphism. We determine a recurrence relation for calculating this number for
a path of any length. If $T_n$ is the number of configurations with period
length 2 that can exist on $P_n$ up to isomorphism and $n$ is an integer
greater than 4, we conclude that $T_n = 3T_{n-1} + 2T_{n-2} + T_{n-3} -
T_{n-4}$.
