  We study the 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 [P. Di Francesco and R. Soto-Garrido. Arctic curves
of the octahedron equation. J. Phys. A, 47(28):285204, 34, 2014], we consider
initial data along parallel ``slanted" planes perpendicular to an arbitrary
admissible direction $(r,s,t)\in {\mathbb Z}_+^3$. The solution of the T-system
is interpreted as the partition function of a dimer model on some suitable
``pinecone" graph introduced in [M. Bousquet-M\'elou, J. Propp, and J. West.
Perfect matchings for the three-term Gale-Robinson sequences. Electron. J.
Combin., 16(1):Research Paper 125, 37, 2009]. 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.
