  We introduce Z-Sequence, a novel empirical model that utilises photometric
measurements of observed galaxies within a specified search radius to estimate
the photometric redshift of galaxy clusters. Z-Sequence itself is composed of a
machine learning ensemble based on the k-nearest neighbours algorithm. We
implement an automated feature selection strategy that iteratively determines
appropriate combinations of filters and colours to minimise photometric
redshift prediction error. We intend for Z-Sequence to be a standalone
technique but it can be combined with cluster finders that do not intrinsically
predict redshift, such as our own DEEP-CEE. In this proof-of-concept study we
train, fine-tune and test Z-Sequence on publicly available cluster catalogues
derived from the Sloan Digital Sky Survey. We determine the photometric
redshift prediction error of Z-Sequence via the median value of $|\Delta
z|/(1+z)$ (across a photometric redshift range of $0.05 \le \textit{z} \le
0.6$) to be $\sim0.01$ when applying a small search radius. The photometric
redshift prediction error for test samples increases by 30-50 per cent when the
search radius is enlarged, likely due to line-of-sight interloping galaxies.
Eventually, we aim to apply Z-Sequence to upcoming imaging surveys such as the
Legacy Survey of Space and Time to provide photometric redshift estimates for
large samples of as yet undiscovered and distant clusters.
