T-system of type A_\infty, also known as the octahedron recurrence/equation, viewed as a 2+1-dimensional discrete evolution equation. Generalizing the study of Di Francesco and Soto-Garrido, we consider initial data along parallel ``slanted" planes perpendicular to an arbitrary admissible direction (r,s,t)?Z3+. The solution of the T-system is interpreted as the partition function of a dimer model on some suitable ``pinecone" graph introduced in Bousquet-Mélou, J. Propp, and J. West. The T-system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system.
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