For Sierpinski polygon graphs with r not a multiple of 4, isomorphism is classified by a dihedral group action on the defining sequence, and eventually-constant sequences yield exactly two Busemann and countably many non-Busemann horofunctions.
Horofunctions on sierpinski type triangles
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Horofunctions of infinite Sierpinski polygon graphs
For Sierpinski polygon graphs with r not a multiple of 4, isomorphism is classified by a dihedral group action on the defining sequence, and eventually-constant sequences yield exactly two Busemann and countably many non-Busemann horofunctions.