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We study cluster algebras arising from cluster tubes. We obtain categoricalinterpretations for $g$-vectors, $c$-vectors and denominator vectors forcluster algebras of type $\mathrm{C}$ with respect to arbitrary initial seeds.In particular, a denominator theorem has been proved, which enables us toestablish the linearly independence of denominator vectors of cluster variablesfrom the same cluster for cluster algebras of type$\mathrm{A}\mathrm{B}\mathrm{C}$. This strengthens the link between clustertubes and cluster algebras of type $\mathrm{C}$ initiated by Buan, Marsh andVatne.

Cluster algebras arising from cluster tubes I: integer vectors